# 4.3 MCQs-Indefinite and Definite Integration

### Indefinite and Definite Integration MCQs

### Antiderivatives and Indefinite Integrals

1\. The indefinite integral $$\int f(x) , dx$$ represents:

1. The area under the curve $$f(x)$$
2. The derivative of $$f(x)$$
3. The family of all antiderivatives of $$f(x)$$
4. The slope of the tangent line to $$f(x)$$

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**Answer:** 3. The family of all antiderivatives of $$f(x)$$

**Explanation:**

* If $$F'(x) = f(x)$$, then $$F(x)$$ is an antiderivative of $$f(x)$$.
* The **indefinite integral** $$\int f(x) , dx = F(x) + C$$, where C is the constant of integration, represents the entire family of antiderivatives.
* Each member of this family differs by a constant.

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2\. The Power Rule for integration states that for $$n \neq -1$$, $$\int x^n , dx =$$

1. $$\frac{x^{n+1}}{n+1} + C$$
2. $$\frac{x^{n-1}}{n-1} + C$$
3. $$n x^{n-1} + C$$
4. $$x^{n+1} + C$$

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**Answer:** 1. $$\frac{x^{n+1}}{n+1} + C$$

**Explanation:**

* This is the reverse of the Power Rule for differentiation: $$\frac{d}{dx}\left( \frac{x^{n+1}}{n+1} \right) = x^n$$.
* The condition $$n \neq -1$$ is important because it leads to the integral $$\int x^{-1} dx = \int \frac{1}{x} dx = \ln|x| + C$$.
* Example: $$\int x^3 dx = \frac{x^{4}}{4} + C$$.

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3\. $$\int e^x , dx =$$

1. $$\ln x + C$$
2. $$x e^{x-1} + C$$
3. $$e^x + C$$
4. $$\frac{e^{x+1}}{x+1} + C$$

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**Answer:** 3. $$e^x + C$$

**Explanation:**

* Since the derivative of $$e^x$$ is $$e^x$$, its antiderivative is also $$e^x$$, plus the constant of integration.
* This is a unique and important property of the exponential function with base $$e$$.

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4\. $$\int \frac{1}{x} , dx =$$

1. $$\ln x + C$$
2. $$\ln |x| + C$$
3. $$\frac{1}{x^2} + C$$
4. $$x^{-1} + C$$

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**Answer:** 2. $$\ln |x| + C$$

**Explanation:**

* The absolute value is crucial because the domain of $$\frac{1}{x}$$ excludes $$x=0$$, and the antiderivative $$\ln x$$ is only defined for $$x > 0$$.
* For $$x < 0$$, the derivative of $$\ln(-x)$$ is also $$\frac{1}{x}$$.
* Therefore, the general antiderivative is $$\ln |x| + C$$, valid for all $$x \neq 0$$.

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### Basic Integration Rules and Substitution

5\. The Constant Multiple Rule for integration states: $$\int k \cdot f(x) , dx =$$

1. $$k + \int f(x) , dx$$
2. $$k \cdot \int f(x) , dx$$
3. $$\int k , dx \cdot \int f(x) , dx$$
4. $$f(kx) + C$$

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**Answer:** 2. $$k \cdot \int f(x) , dx$$

**Explanation:**

* Constants can be factored out of the integral. Formally, $$\int k f(x) dx = k \int f(x) dx$$, where k is any constant.
* This follows directly from the linearity of the derivative: the derivative of $$k F(x)$$ is $$k f(x)$$.

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6\. The Sum Rule for integration states: $$\int \[f(x) + g(x)] , dx =$$

1. $$\int f(x) , dx \cdot \int g(x) , dx$$
2. $$\int f(x) , dx + \int g(x) , dx$$
3. $$f(x) + g(x) + C$$
4. $$\frac{1}{2} \int f(x) , dx + \frac{1}{2} \int g(x) , dx$$

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**Answer:** 2. $$\int f(x) , dx + \int g(x) , dx$$

**Explanation:**

* The integral of a sum is the sum of the integrals. Formally, $$\int \[f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$$.
* Combined with the Constant Multiple Rule, this makes integration a **linear operation**.
* Example: $$\int (3x^2 + 2\cos x) dx = \int 3x^2 dx + \int 2\cos x dx = x^3 + 2\sin x + C$$.

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7\. The Substitution Rule (u-substitution) for integration is essentially:

1. The Power Rule in reverse
2. The Chain Rule in reverse
3. The Product Rule in reverse
4. The Quotient Rule in reverse

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**Answer:** 2. The Chain Rule in reverse

**Explanation:**

* If an integral is of the form $$\int f(g(x)) g'(x) dx$$, we can set $$u = g(x)$$, then $$du = g'(x) dx$$.
* The integral then simplifies to $$\int f(u) du$$, which is hopefully easier to evaluate.
* This method "undoes" the Chain Rule for differentiation.
* Example: $$\int 2x \cos(x^2) dx$$. Let $$u = x^2$$, $$du = 2x dx$$. The integral becomes $$\int \cos u , du = \sin u + C = \sin(x^2) + C$$.

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### Definite Integrals and The Fundamental Theorem

8\. The definite integral $$\int\_{a}^{b} f(x) , dx$$ geometrically represents:

1. The slope of the secant line from $$(a, f(a))$$ to $$(b, f(b))$$
2. The average value of $$f(x)$$ on $$\[a, b]$$
3. The net signed area between the curve $$y=f(x)$$, the x-axis, and the lines $$x=a$$ and $$x=b$$
4. The derivative of $$f(x)$$ at $$x=b$$

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**Answer:** 3. The net signed area between the curve $$y=f(x)$$, the x-axis, and the lines $$x=a$$ and $$x=b$$

**Explanation:**

* Area **above** the x-axis is counted positively.
* Area **below** the x-axis is counted negatively.
* The result is the **net** signed area.
* This is the primary geometric interpretation of the definite integral.

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9\. The First Fundamental Theorem of Calculus states that if $$f$$ is continuous on $$\[a, b]$$ and $$F$$ is an antiderivative of $$f$$ (i.e., $$F' = f$$), then:

1. $$\frac{d}{dx} \int\_{a}^{x} f(t) dt = f(x)$$
2. $$\int\_{a}^{b} f(x) dx = F(b) - F(a)$$
3. Both 1 and 2
4. $$\int\_{a}^{b} F(x) dx = f(b) - f(a)$$

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**Answer:** 3. Both 1 and 2

**Explanation:**

* The First Fundamental Theorem of Calculus has two closely related parts.
* **Part 1:** If $$g(x) = \int\_{a}^{x} f(t) dt$$, then $$g'(x) = f(x)$$. This links differentiation and integration.
* **Part 2 (Evaluation Theorem):** $$\int\_{a}^{b} f(x) dx = F(b) - F(a)$$. This provides a practical way to evaluate definite integrals using antiderivatives.
* Notation: $$F(b) - F(a)$$ is often written as $$\left\[ F(x) \right]*{a}^{b}$$ or $$F(x) \big|*{a}^{b}$$.

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10\. Using the Evaluation Theorem, $$\int\_{1}^{3} 2x , dx =$$

1. 4
2. 8
3. 9
4. 10

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**Answer:** 2. 8

**Explanation:**

* Find an antiderivative: $$\int 2x dx = x^2 + C$$. We can use $$F(x) = x^2$$.
* Apply the theorem: $$\int\_{1}^{3} 2x dx = F(3) - F(1) = (3)^2 - (1)^2 = 9 - 1 = 8$$.
* Geometrically, this is the area of a trapezoid: base from 1 to 3, with heights 2 and 6. Area = average height (4) × width (2) = 8.

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### Properties of Definite Integrals

11\. The property $$\int\_{a}^{b} f(x) , dx = -\int\_{b}^{a} f(x) , dx$$ reflects that:

1. Reversing the limits of integration changes the sign of the integral.
2. The integral is symmetric.
3. The area is always positive.
4. The function is odd.

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**Answer:** 1. Reversing the limits of integration changes the sign of the integral.

**Explanation:**

* This is a definition/convention that ensures consistency with the Evaluation Theorem.
* If $$F$$ is an antiderivative, then $$\int\_{a}^{b} f(x) dx = F(b)-F(a)$$.
* Conversely, $$\int\_{b}^{a} f(x) dx = F(a)-F(b) = -(F(b)-F(a)) = -\int\_{a}^{b} f(x) dx$$.

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12\. The Additivity property for definite integrals states that for $$a < c < b$$:

1. $$\int\_{a}^{b} f(x) dx = \int\_{a}^{c} f(x) dx + \int\_{c}^{b} f(x) dx$$
2. $$\int\_{a}^{b} f(x) dx = \int\_{a}^{c} f(x) dx \cdot \int\_{c}^{b} f(x) dx$$
3. $$\int\_{a}^{b} f(x) dx = \int\_{a}^{b} \[f(x)+g(x)] dx$$
4. $$\int\_{a}^{b} f(x) dx = \int\_{c}^{d} f(x) dx$$

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**Answer:** 1. $$\int\_{a}^{b} f(x) dx = \int\_{a}^{c} f(x) dx + \int\_{c}^{b} f(x) dx$$

**Explanation:**

* This property is intuitive from the area interpretation: The total area from a to b is the sum of the area from a to c and the area from c to b.
* This property is very useful for splitting integrals, especially when a function is defined piecewise or has different behaviors on different intervals.

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### Integration by Parts

13\. The formula for Integration by Parts is derived from:

1. The Chain Rule
2. The Product Rule
3. The Quotient Rule
4. The Power Rule

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**Answer:** 2. The Product Rule

**Explanation:**

* Starting from the Product Rule: $$\frac{d}{dx}\[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$.
* Integrate both sides: $$u(x)v(x) = \int u'(x)v(x) dx + \int u(x)v'(x) dx$$.
* Rearrange: $$\int u(x)v'(x) dx = u(x)v(x) - \int u'(x)v(x) dx$$.
* In differential form (letting $$dv = v'(x)dx$$ and $$du = u'(x)dx$$): $$\int u , dv = uv - \int v , du$$.

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14\. Integration by Parts is most useful for integrals of the form:

1. $$\int \sin(x^2) dx$$
2. $$\int x e^x dx$$
3. $$\int e^{x^2} dx$$
4. $$\int \frac{\ln x}{x} dx$$

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**Answer:** 2. $$\int x e^x dx$$

**Explanation:**

* Integration by Parts is effective for integrals that are products of different types of functions, such as polynomial × exponential ($$x^n e^x$$), polynomial × trigonometric ($$x^n \sin x$$), or logarithmic × polynomial ($$\ln x \cdot x^n$$).
* For $$\int x e^x dx$$, a good choice is: $$u = x$$ (so $$du = dx$$) and $$dv = e^x dx$$ (so $$v = e^x$$).
* Then $$\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C = e^x(x-1) + C$$.
* Options 1 and 3 do not have elementary antiderivatives. Option 4 is better solved by simple substitution (let $$u = \ln x$$).

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