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:

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

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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:

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

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

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

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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 $x = a\sec\theta$, $dx = a\sec\theta\tan\theta d\theta$

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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:

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

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

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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 $b^2-4ac < 0$:

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

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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:

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

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

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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 $[a,b]$.

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

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

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

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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 $x=4$, $u=10$

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

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

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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.

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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 $0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c$

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

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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

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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. $\int \cot x dx = \ln|\sin x| + C$

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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 $x=1$, $u=0$

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

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

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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.