# 4.3 Indefinite and definite Integration

## Detailed Theory: Indefinite and Definite Integration

### **1. Introduction to Integration**

#### **1.1 What is Integration?**

Integration is the reverse process of differentiation. If differentiation gives us the rate of change, integration gives us the accumulation of quantities.

**Two main types:**

1. **Indefinite Integration:** Finds antiderivatives (general form)
2. **Definite Integration:** Computes accumulated quantity over an interval

#### **1.2 The Integral Symbol**

The integral symbol $$\int$$ comes from the elongated "S" meaning "sum."

* Indefinite integral: $$\int f(x) dx = F(x) + C$$
* Definite integral: $$\int\_{a}^{b} f(x) dx = \[F(x)]\_{a}^{b} = F(b) - F(a)$$

#### **1.3 Relationship with Derivatives**

If $$F'(x) = f(x)$$, then:

$$
\int f(x) dx = F(x) + C
$$

where $$C$$ is the constant of integration.

**Example:** Since $$\frac{d}{dx}(x^3) = 3x^2$$, then:

$$
\int 3x^2 dx = x^3 + C
$$

***

### **2. Indefinite Integration**

#### **2.1 Basic Integration Formulas**

**Power Rule (for n ≠ -1)**

$$
\int x^n dx = \frac{x^{n+1}}{n+1} + C
$$

**Examples:**

* $$\int x^3 dx = \frac{x^4}{4} + C$$
* $$\int \sqrt{x} dx = \int x^{1/2} dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$$
* $$\int \frac{1}{x^2} dx = \int x^{-2} dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$$

**Exponential Functions**

1. $$\int e^x dx = e^x + C$$
2. $$\int a^x dx = \frac{a^x}{\ln a} + C$$ for $$a > 0$$, $$a \neq 1$$

**Trigonometric Functions**

1. $$\int \sin x dx = -\cos x + C$$
2. $$\int \cos x dx = \sin x + C$$
3. $$\int \sec^2 x dx = \tan x + C$$
4. $$\int \csc^2 x dx = -\cot x + C$$
5. $$\int \sec x \tan x dx = \sec x + C$$
6. $$\int \csc x \cot x dx = -\csc x + C$$

**Inverse Trigonometric Forms**

1. $$\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + C$$
2. $$\int \frac{1}{1+x^2} dx = \tan^{-1} x + C$$
3. $$\int \frac{1}{|x|\sqrt{x^2-1}} dx = \sec^{-1} x + C$$

#### **2.2 Properties of Indefinite Integrals**

**Linearity Properties**

1. $$\int \[f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$$
2. $$\int \[f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx$$
3. $$\int k f(x) dx = k \int f(x) dx$$ for constant $$k$$

**Constant Multiple Rule**

For constant $$c$$:

$$
\int c f(x) dx = c \int f(x) dx
$$

**Sum/Difference Rule**

$$
\int \[f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx
$$

#### **2.3 Example: Using Basic Rules**

Find $$\int (3x^2 - 2x + 5) dx$$

**Solution:**

$$
\int (3x^2 - 2x + 5) dx = 3\int x^2 dx - 2\int x dx + 5\int dx
$$

$$
\= 3 \cdot \frac{x^3}{3} - 2 \cdot \frac{x^2}{2} + 5x + C
$$

$$
\= x^3 - x^2 + 5x + C
$$

***

### **3. Integration by Substitution**

#### **3.1 The Method**

Substitution is the reverse of chain rule. For integral $$\int f(g(x))g'(x) dx$$:

Let $$u = g(x)$$, then $$du = g'(x) dx$$

The integral becomes: $$\int f(u) du$$

#### **3.2 Steps for Substitution**

1. Choose substitution $$u = g(x)$$
2. Compute $$du = g'(x) dx$$
3. Rewrite integral in terms of $$u$$
4. Integrate with respect to $$u$$
5. Substitute back $$u = g(x)$$

#### **3.3 Examples**

**Example 1:** Find $$\int 2x(x^2 + 1)^3 dx$$

Let $$u = x^2 + 1$$, then $$du = 2x dx$$

The integral becomes: $$\int u^3 du = \frac{u^4}{4} + C$$

Substitute back: $$\frac{(x^2 + 1)^4}{4} + C$$

**Example 2:** Find $$\int \sin(3x) dx$$

Let $$u = 3x$$, then $$du = 3 dx$$, so $$dx = \frac{du}{3}$$

$$
\int \sin(3x) dx = \int \sin u \cdot \frac{du}{3} = \frac{1}{3} \int \sin u du
$$

$$
\= -\frac{1}{3} \cos u + C = -\frac{1}{3} \cos(3x) + C
$$

**Example 3:** Find $$\int x e^{x^2} dx$$

Let $$u = x^2$$, then $$du = 2x dx$$, so $$x dx = \frac{du}{2}$$

$$
\int x e^{x^2} dx = \int e^u \cdot \frac{du}{2} = \frac{1}{2} \int e^u du
$$

$$
\= \frac{1}{2} e^u + C = \frac{1}{2} e^{x^2} + C
$$

#### **3.4 Trigonometric Substitution**

Used for integrals containing $$\sqrt{a^2 - x^2}$$, $$\sqrt{a^2 + x^2}$$, or $$\sqrt{x^2 - a^2}$$.

**Three main cases:**

1. **For** $$\sqrt{a^2 - x^2}$$**:** Use $$x = a\sin\theta$$, $$dx = a\cos\theta d\theta$$

   Example: $$\int \frac{dx}{\sqrt{4-x^2}}$$

   Let $$x = 2\sin\theta$$, $$dx = 2\cos\theta d\theta$$

   $$
   \int \frac{2\cos\theta d\theta}{\sqrt{4-4\sin^2\theta}} = \int \frac{2\cos\theta d\theta}{2\cos\theta} = \int d\theta = \theta + C
   $$

   Back substitute: $$\theta = \sin^{-1}\left(\frac{x}{2}\right)$$

   So $$\int \frac{dx}{\sqrt{4-x^2}} = \sin^{-1}\left(\frac{x}{2}\right) + C$$
2. **For** $$\sqrt{a^2 + x^2}$$**:** Use $$x = a\tan\theta$$, $$dx = a\sec^2\theta d\theta$$
3. **For** $$\sqrt{x^2 - a^2}$$**:** Use $$x = a\sec\theta$$, $$dx = a\sec\theta\tan\theta d\theta$$

***

### **4. Integration by Parts**

#### **4.1 Formula**

Based on product rule: $$\frac{d}{dx}(uv) = u'v + uv'$$

Integration gives: $$\int u dv = uv - \int v du$$

#### **4.2 Choosing u and dv**

Use **LIATE** rule for choosing $$u$$ (in order of preference):

* **L:** Logarithmic functions ($$\ln x$$)
* **I:** Inverse trigonometric functions
* **A:** Algebraic functions (polynomials)
* **T:** Trigonometric functions
* **E:** Exponential functions

#### **4.3 Examples**

**Example 1:** Find $$\int x e^x dx$$

Let $$u = x$$, $$dv = e^x dx$$ Then $$du = dx$$, $$v = e^x$$

$$
\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C = e^x(x-1) + C
$$

**Example 2:** Find $$\int \ln x dx$$

Let $$u = \ln x$$, $$dv = dx$$ Then $$du = \frac{1}{x} dx$$, $$v = x$$

$$
\int \ln x dx = x\ln x - \int x \cdot \frac{1}{x} dx = x\ln x - \int dx = x\ln x - x + C
$$

**Example 3:** Find $$\int e^x \sin x dx$$

Let $$u = e^x$$, $$dv = \sin x dx$$ Then $$du = e^x dx$$, $$v = -\cos x$$

$$
\int e^x \sin x dx = -e^x \cos x + \int e^x \cos x dx
$$

Apply integration by parts again to $$\int e^x \cos x dx$$: Let $$u = e^x$$, $$dv = \cos x dx$$ Then $$du = e^x dx$$, $$v = \sin x$$

$$
\int e^x \cos x dx = e^x \sin x - \int e^x \sin x dx
$$

Now we have:

$$
\int e^x \sin x dx = -e^x \cos x + e^x \sin x - \int e^x \sin x dx
$$

Bring the integral to left side:

$$
2\int e^x \sin x dx = e^x(\sin x - \cos x)
$$

$$
\int e^x \sin x dx = \frac{e^x}{2}(\sin x - \cos x) + C
$$

***

### **5. Integration by Partial Fractions**

#### **5.1 When to Use**

For rational functions $$\frac{P(x)}{Q(x)}$$ where degree of $$P(x)$$ < degree of $$Q(x)$$.

#### **5.2 Method**

1. Factor denominator $$Q(x)$$
2. Write as sum of partial fractions
3. Solve for unknown coefficients
4. Integrate each term

#### **5.3 Cases for Partial Fractions**

**Case 1: Distinct Linear Factors**

For $$\frac{P(x)}{(x-a)(x-b)}$$:

$$
\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}
$$

**Example:** $$\int \frac{dx}{x^2 - 1}$$

Factor: $$x^2 - 1 = (x-1)(x+1)$$

Write: $$\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$$

Multiply through: $$1 = A(x+1) + B(x-1)$$

Solve: Let $$x=1$$: $$1 = A(2) \Rightarrow A = \frac{1}{2}$$

Let $$x=-1$$: $$1 = B(-2) \Rightarrow B = -\frac{1}{2}$$

So: $$\int \frac{dx}{x^2-1} = \int \left(\frac{1/2}{x-1} - \frac{1/2}{x+1}\right) dx$$

$$
\= \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C
$$

**Case 2: Repeated Linear Factors**

For $$\frac{P(x)}{(x-a)^n}$$:

$$
\frac{P(x)}{(x-a)^n} = \frac{A\_1}{x-a} + \frac{A\_2}{(x-a)^2} + \cdots + \frac{A\_n}{(x-a)^n}
$$

**Case 3: Irreducible Quadratic Factors**

For $$\frac{P(x)}{(ax^2+bx+c)}$$ with $$b^2-4ac < 0$$:

$$
\frac{P(x)}{ax^2+bx+c} = \frac{Ax+B}{ax^2+bx+c}
$$

***

### **6. Trigonometric Integrals**

#### **6.1 Integrals of Powers of sin and cos**

**Case 1:** $$\int \sin^m x \cos^n x dx$$

**Strategy:**

* If $$m$$ is odd: Save one $$\sin x$$, use $$\sin^2 x = 1 - \cos^2 x$$
* If $$n$$ is odd: Save one $$\cos x$$, use $$\cos^2 x = 1 - \sin^2 x$$
* If both even: Use power reduction formulas

**Example:** $$\int \sin^3 x \cos^2 x dx$$

Here $$m=3$$ (odd), $$n=2$$ (even)

$$
\int \sin^3 x \cos^2 x dx = \int \sin^2 x \cos^2 x \sin x dx
$$

$$
\= \int (1-\cos^2 x) \cos^2 x \sin x dx
$$

Let $$u = \cos x$$, $$du = -\sin x dx$$

$$
\= -\int (1-u^2)u^2 du = -\int (u^2 - u^4) du
$$

$$
\= -\left(\frac{u^3}{3} - \frac{u^5}{5}\right) + C = -\frac{\cos^3 x}{3} + \frac{\cos^5 x}{5} + C
$$

#### **6.2 Integrals of Powers of tan and sec**

**Case 2:** $$\int \tan^m x \sec^n x dx$$

**Strategy:**

* If $$n$$ is even: Save $$\sec^2 x$$, use $$\sec^2 x = 1 + \tan^2 x$$
* If $$m$$ is odd: Save $$\sec x \tan x$$, use $$\tan^2 x = \sec^2 x - 1$$

#### **6.3 Power Reduction Formulas**

Useful for even powers:

$$
\sin^2 x = \frac{1 - \cos 2x}{2}
$$

$$
\cos^2 x = \frac{1 + \cos 2x}{2}
$$

**Example:** $$\int \sin^2 x dx$$

Using formula:

$$
\int \sin^2 x dx = \int \frac{1 - \cos 2x}{2} dx = \frac{1}{2}\int dx - \frac{1}{2}\int \cos 2x dx
$$

$$
\= \frac{1}{2}x - \frac{1}{4}\sin 2x + C
$$

***

### **7. Definite Integrals**

#### **7.1 Definition: Riemann Sum**

The definite integral $$\int\_{a}^{b} f(x) dx$$ is defined as:

$$
\int\_{a}^{b} f(x) dx = \lim\_{n \to \infty} \sum\_{i=1}^{n} f(x\_i^\*) \Delta x
$$

where $$\Delta x = \frac{b-a}{n}$$ and $$x\_i^\*$$ is a point in the i-th subinterval.

#### **7.2 Properties of Definite Integrals**

**Basic Properties**

1. $$\int\_{a}^{a} f(x) dx = 0$$
2. $$\int\_{a}^{b} f(x) dx = -\int\_{b}^{a} f(x) dx$$
3. $$\int\_{a}^{b} \[f(x) + g(x)] dx = \int\_{a}^{b} f(x) dx + \int\_{a}^{b} g(x) dx$$
4. $$\int\_{a}^{b} cf(x) dx = c\int\_{a}^{b} f(x) dx$$
5. $$\int\_{a}^{b} f(x) dx = \int\_{a}^{c} f(x) dx + \int\_{c}^{b} f(x) dx$$ for any $$c$$

**Comparison Properties**

1. If $$f(x) \geq 0$$ on $$\[a,b]$$, then $$\int\_{a}^{b} f(x) dx \geq 0$$
2. If $$f(x) \geq g(x)$$ on $$\[a,b]$$, then $$\int\_{a}^{b} f(x) dx \geq \int\_{a}^{b} g(x) dx$$
3. If $$m \leq f(x) \leq M$$ on $$\[a,b]$$, then:

$$
m(b-a) \leq \int\_{a}^{b} f(x) dx \leq M(b-a)
$$

#### **7.3 Mean Value Theorem for Integrals**

If $$f$$ is continuous on $$\[a,b]$$, then there exists $$c$$ in $$\[a,b]$$ such that:

$$
f(c) = \frac{1}{b-a} \int\_{a}^{b} f(x) dx
$$

This value $$f(c)$$ is called the **average value** of $$f$$ on $$\[a,b]$$.

***

### **8. Fundamental Theorem of Calculus**

#### **8.1 Part 1**

If $$f$$ is continuous on $$\[a,b]$$ and $$F(x) = \int\_{a}^{x} f(t) dt$$, then:

$$
F'(x) = f(x) \quad \text{for } a < x < b
$$

**In words:** Derivative of integral with variable upper limit = integrand evaluated at upper limit.

**Example:** If $$F(x) = \int\_{0}^{x} \sin(t^2) dt$$, then $$F'(x) = \sin(x^2)$$

#### **8.2 Part 2 (Evaluation Theorem)**

If $$f$$ is continuous on $$\[a,b]$$ and $$F$$ is any antiderivative of $$f$$, then:

$$
\int\_{a}^{b} f(x) dx = F(b) - F(a)
$$

**Notation:** $$\[F(x)]\_{a}^{b} = F(b) - F(a)$$

**Example:** Evaluate $$\int\_{1}^{2} x^2 dx$$

Antiderivative: $$F(x) = \frac{x^3}{3}$$

$$
\int\_{1}^{2} x^2 dx = \left\[\frac{x^3}{3}\right]\_{1}^{2} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}
$$

#### **8.3 Examples with FTC**

**Example 1:** Find derivative of $$g(x) = \int\_{0}^{x^2} \cos(t) dt$$

Let $$u = x^2$$, then $$g(x) = \int\_{0}^{u} \cos(t) dt$$

By Chain Rule and FTC Part 1:

$$
g'(x) = \cos(u) \cdot \frac{du}{dx} = \cos(x^2) \cdot 2x = 2x \cos(x^2)
$$

**Example 2:** Evaluate $$\int\_{0}^{\pi} \sin x dx$$

Antiderivative: $$F(x) = -\cos x$$

$$
\int\_{0}^{\pi} \sin x dx = \[-\cos x]\_{0}^{\pi} = (-\cos \pi) - (-\cos 0)
$$

$$
\= (-(-1)) - (-1) = 1 + 1 = 2
$$

***

### **9. Applications of Definite Integrals**

#### **9.1 Area Under a Curve**

Area between $$y = f(x)$$ and x-axis from $$x = a$$ to $$x = b$$:

$$
A = \int\_{a}^{b} |f(x)| dx
$$

**Example:** Area under $$y = x^2$$ from $$x = 0$$ to $$x = 2$$

$$
A = \int\_{0}^{2} x^2 dx = \left\[\frac{x^3}{3}\right]\_{0}^{2} = \frac{8}{3}
$$

#### **9.2 Area Between Two Curves**

Area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a$$ to $$x = b$$ (where $$f(x) \geq g(x)$$):

$$
A = \int\_{a}^{b} \[f(x) - g(x)] dx
$$

**Example:** Area between $$y = x^2$$ and $$y = x$$ from $$x = 0$$ to $$x = 1$$

Find intersection: $$x^2 = x \Rightarrow x(x-1) = 0 \Rightarrow x=0,1$$

Between 0 and 1, $$x \geq x^2$$

$$
A = \int\_{0}^{1} (x - x^2) dx = \left\[\frac{x^2}{2} - \frac{x^3}{3}\right]\_{0}^{1} = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}
$$

#### **9.3 Volume by Slicing (Disk/Washer Method)**

**Disk Method (rotation around x-axis)**

Volume when region under $$y = f(x)$$ rotated about x-axis:

$$
V = \pi \int\_{a}^{b} \[f(x)]^2 dx
$$

**Example:** Volume of solid formed by rotating $$y = \sqrt{x}$$ from $$x=0$$ to $$x=4$$ about x-axis

$$
V = \pi \int\_{0}^{4} (\sqrt{x})^2 dx = \pi \int\_{0}^{4} x dx = \pi \left\[\frac{x^2}{2}\right]\_{0}^{4} = \pi \cdot 8 = 8\pi
$$

**Washer Method**

When there's a hole: $$V = \pi \int\_{a}^{b} (\[R(x)]^2 - \[r(x)]^2) dx$$ where $$R(x)$$ = outer radius, $$r(x)$$ = inner radius

#### **9.4 Arc Length**

Length of curve $$y = f(x)$$ from $$x = a$$ to $$x = b$$:

$$
L = \int\_{a}^{b} \sqrt{1 + \[f'(x)]^2} dx
$$

**Example:** Arc length of $$y = x^{3/2}$$ from $$x=0$$ to $$x=4$$

$$f'(x) = \frac{3}{2}x^{1/2}$$, so $$\[f'(x)]^2 = \frac{9}{4}x$$

$$
L = \int\_{0}^{4} \sqrt{1 + \frac{9}{4}x} dx
$$

Let $$u = 1 + \frac{9}{4}x$$, $$du = \frac{9}{4}dx$$

When $$x=0$$, $$u=1$$; when $$x=4$$, $$u=10$$

$$
L = \int\_{1}^{10} \sqrt{u} \cdot \frac{4}{9} du = \frac{4}{9} \cdot \frac{2}{3}\[u^{3/2}]\_{1}^{10}
$$

$$
\= \frac{8}{27}(10^{3/2} - 1^{3/2}) = \frac{8}{27}(10\sqrt{10} - 1)
$$

***

### **10. Improper Integrals**

#### **10.1 Types of Improper Integrals**

**Type 1: Infinite Intervals**

1. $$\int\_{a}^{\infty} f(x) dx = \lim\_{t \to \infty} \int\_{a}^{t} f(x) dx$$
2. $$\int\_{-\infty}^{b} f(x) dx = \lim\_{t \to -\infty} \int\_{t}^{b} f(x) dx$$
3. $$\int\_{-\infty}^{\infty} f(x) dx = \int\_{-\infty}^{c} f(x) dx + \int\_{c}^{\infty} f(x) dx$$

**Type 2: Discontinuous Integrands**

If $$f$$ is discontinuous at $$a$$:

$$
\int\_{a}^{b} f(x) dx = \lim\_{t \to a^+} \int\_{t}^{b} f(x) dx
$$

If $$f$$ is discontinuous at $$b$$:

$$
\int\_{a}^{b} f(x) dx = \lim\_{t \to b^-} \int\_{a}^{t} f(x) dx
$$

#### **10.2 Convergence Tests**

**Comparison Test**

If $$0 \leq f(x) \leq g(x)$$ for $$x \geq a$$:

* If $$\int\_{a}^{\infty} g(x) dx$$ converges, then $$\int\_{a}^{\infty} f(x) dx$$ converges
* If $$\int\_{a}^{\infty} f(x) dx$$ diverges, then $$\int\_{a}^{\infty} g(x) dx$$ diverges

**p-integral Test**

$$
\int\_{1}^{\infty} \frac{1}{x^p} dx \quad \begin{cases}
\text{converges if } p > 1 \\
\text{diverges if } p \leq 1
\end{cases}
$$

#### **10.3 Examples**

**Example 1:** $$\int\_{1}^{\infty} \frac{1}{x^2} dx$$

$$
\int\_{1}^{\infty} \frac{1}{x^2} dx = \lim\_{t \to \infty} \int\_{1}^{t} x^{-2} dx = \lim\_{t \to \infty} \left\[-\frac{1}{x}\right]\_{1}^{t}
$$

$$
\= \lim\_{t \to \infty} \left(-\frac{1}{t} + 1\right) = 1
$$

Converges to 1.

**Example 2:** $$\int\_{0}^{1} \frac{1}{\sqrt{x}} dx$$

Discontinuity at $$x=0$$

$$
\int\_{0}^{1} \frac{1}{\sqrt{x}} dx = \lim\_{t \to 0^+} \int\_{t}^{1} x^{-1/2} dx = \lim\_{t \to 0^+} \[2\sqrt{x}]\_{t}^{1}
$$

$$
\= \lim\_{t \to 0^+} (2\sqrt{1} - 2\sqrt{t}) = 2
$$

Converges to 2.

***

### **11. Multiple Integration**

#### **11.1 Double Integrals**

For function $$f(x,y)$$ over region $$R$$:

$$
\iint\_{R} f(x,y) dA
$$

#### **11.2 Iterated Integrals**

**Type I Region (vertically simple):**

$$
\iint\_{R} f(x,y) dA = \int\_{a}^{b} \left\[\int\_{g\_1(x)}^{g\_2(x)} f(x,y) dy\right] dx
$$

**Type II Region (horizontally simple):**

$$
\iint\_{R} f(x,y) dA = \int\_{c}^{d} \left\[\int\_{h\_1(y)}^{h\_2(y)} f(x,y) dx\right] dy
$$

#### **11.3 Example: Double Integral**

Evaluate $$\iint\_{R} (x + 2y) dA$$ where $$R = {(x,y): 0 \leq x \leq 2, 0 \leq y \leq 1}$$

$$
\iint\_{R} (x + 2y) dA = \int\_{0}^{2} \int\_{0}^{1} (x + 2y) dy dx
$$

First integrate with respect to $$y$$:

$$
\int\_{0}^{1} (x + 2y) dy = \left\[xy + y^2\right]\_{0}^{1} = x + 1
$$

Then with respect to $$x$$:

$$
\int\_{0}^{2} (x + 1) dx = \left\[\frac{x^2}{2} + x\right]\_{0}^{2} = (2 + 2) - 0 = 4
$$

#### **11.4 Triple Integrals**

For function $$f(x,y,z)$$ over solid $$E$$:

$$
\iiint\_{E} f(x,y,z) dV
$$

**Example:** Volume of box $$0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c$$

$$
V = \int\_{0}^{a} \int\_{0}^{b} \int\_{0}^{c} dz dy dx = abc
$$

***

### **12. Numerical Integration**

#### **12.1 When Numerical Methods are Used**

* When antiderivative cannot be found in elementary functions
* When only data points are available, not function formula
* For quick approximations

#### **12.2 Midpoint Rule**

Approximate $$\int\_{a}^{b} f(x) dx$$ using midpoints of subintervals:

$$
M\_n = \Delta x \sum\_{i=1}^{n} f(\bar{x}\_i)
$$

where $$\Delta x = \frac{b-a}{n}$$, $$\bar{x}*i = \frac{x*{i-1} + x\_i}{2}$$

#### **12.3 Trapezoidal Rule**

$$
T\_n = \frac{\Delta x}{2}\[f(x\_0) + 2f(x\_1) + 2f(x\_2) + \cdots + 2f(x\_{n-1}) + f(x\_n)]
$$

where $$\Delta x = \frac{b-a}{n}$$, $$x\_i = a + i\Delta x$$

#### **12.4 Simpson's Rule** (n must be even)

$$
S\_n = \frac{\Delta x}{3}\[f(x\_0) + 4f(x\_1) + 2f(x\_2) + 4f(x\_3) + \cdots + 4f(x\_{n-1}) + f(x\_n)]
$$

**Error Bounds:**

* Trapezoidal: $$|E\_T| \leq \frac{K(b-a)^3}{12n^2}$$
* Simpson's: $$|E\_S| \leq \frac{K(b-a)^5}{180n^4}$$

where $$|f''(x)| \leq K$$ for Trapezoidal, $$|f^{(4)}(x)| \leq K$$ for Simpson's

***

### **13. Important Integration Formulas Summary**

#### **13.1 Basic Integration Formulas**

1. $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ (n ≠ -1)
2. $$\int \frac{1}{x} dx = \ln|x| + C$$
3. $$\int e^x dx = e^x + C$$
4. $$\int \sin x dx = -\cos x + C$$
5. $$\int \cos x dx = \sin x + C$$

#### **13.2 Substitution Rule**

$$\int f(g(x))g'(x) dx = \int f(u) du$$ where $$u = g(x)$$

#### **13.3 Integration by Parts**

$$\int u dv = uv - \int v du$$

#### **13.4 Fundamental Theorem of Calculus**

$$\int\_{a}^{b} f(x) dx = F(b) - F(a)$$ where $$F'(x) = f(x)$$

#### **13.5 Common Trigonometric Integrals**

1. $$\int \tan x dx = \ln|\sec x| + C$$
2. $$\int \sec x dx = \ln|\sec x + \tan x| + C$$
3. $$\int \csc x dx = \ln|\csc x - \cot x| + C$$
4. $$\int \cot x dx = \ln|\sin x| + C$$

***

### **14. Solved Examples**

#### **Example 1:** Integration by Substitution

Find $$\int \frac{x}{\sqrt{1-x^4}} dx$$

**Solution:** Let $$u = x^2$$, then $$du = 2x dx$$, so $$x dx = \frac{du}{2}$$

$$
\int \frac{x}{\sqrt{1-x^4}} dx = \int \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{2} = \frac{1}{2} \sin^{-1} u + C
$$

$$
\= \frac{1}{2} \sin^{-1}(x^2) + C
$$

#### **Example 2:** Integration by Parts

Find $$\int x^2 \ln x dx$$

**Solution:** Let $$u = \ln x$$, $$dv = x^2 dx$$ Then $$du = \frac{1}{x} dx$$, $$v = \frac{x^3}{3}$$

$$
\int x^2 \ln x dx = \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} dx
$$

$$
\= \frac{x^3}{3} \ln x - \frac{1}{3} \int x^2 dx = \frac{x^3}{3} \ln x - \frac{1}{3} \cdot \frac{x^3}{3} + C
$$

$$
\= \frac{x^3}{3} \ln x - \frac{x^3}{9} + C = \frac{x^3}{9}(3\ln x - 1) + C
$$

#### **Example 3:** Partial Fractions

Find $$\int \frac{2x+3}{(x-1)(x+2)} dx$$

**Solution:** Write: $$\frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$

Multiply: $$2x+3 = A(x+2) + B(x-1)$$

Let $$x=1$$: $$5 = A(3) \Rightarrow A = \frac{5}{3}$$

Let $$x=-2$$: $$-1 = B(-3) \Rightarrow B = \frac{1}{3}$$

So:

$$
\int \frac{2x+3}{(x-1)(x+2)} dx = \int \left(\frac{5/3}{x-1} + \frac{1/3}{x+2}\right) dx
$$

$$
\= \frac{5}{3}\ln|x-1| + \frac{1}{3}\ln|x+2| + C
$$

#### **Example 4:** Definite Integral with Substitution

Evaluate $$\int\_{0}^{1} x\sqrt{1-x^2} dx$$

**Solution:** Let $$u = 1-x^2$$, then $$du = -2x dx$$, so $$x dx = -\frac{du}{2}$$

When $$x=0$$, $$u=1$$; when $$x=1$$, $$u=0$$

$$
\int\_{0}^{1} x\sqrt{1-x^2} dx = \int\_{1}^{0} \sqrt{u} \left(-\frac{du}{2}\right) = \frac{1}{2} \int\_{0}^{1} u^{1/2} du
$$

$$
\= \frac{1}{2} \cdot \frac{2}{3}\[u^{3/2}]\_{0}^{1} = \frac{1}{3}(1-0) = \frac{1}{3}
$$

***

### **15. Common Mistakes and Exam Tips**

#### **15.1 Common Mistakes**

1. **Forgetting +C in indefinite integrals**
2. **Misapplying power rule:** $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, not $$\frac{x^{n-1}}{n-1}$$
3. **Chain rule in reverse:** Forgetting to account for derivative of inner function in substitution
4. **Definite integrals:** Forgetting to change limits when using substitution
5. **Partial fractions:** Not checking if degree of numerator < degree of denominator first

#### **15.2 Problem-Solving Strategy**

1. **Identify type:** Which method applies? (substitution, parts, partial fractions, etc.)
2. **Simplify first:** Use algebra/trig identities to simplify integrand
3. **Try substitution:** Often the first method to try
4. **Check answer:** Differentiate your answer to verify
5. **For definite integrals:**
   * Consider symmetry properties
   * Check if function is even/odd
   * Consider splitting interval at discontinuities

#### **15.3 Quick Checks**

1. **Even functions:** $$\int\_{-a}^{a} f(x) dx = 2\int\_{0}^{a} f(x) dx$$ if $$f$$ is even
2. **Odd functions:** $$\int\_{-a}^{a} f(x) dx = 0$$ if $$f$$ is odd
3. **Periodic functions:** $$\int\_{a}^{a+T} f(x) dx = \int\_{0}^{T} f(x) dx$$ if $$f$$ has period $$T$$
4. **Area interpretation:** Definite integral = net area (above x-axis minus below)

This comprehensive theory covers all aspects of indefinite and definite integration with detailed explanations and examples, providing complete preparation for the entrance examination.