Master Integration Techniques: Enhance Your Calculus Skills

Question 1

What is the indefinite integral of ∫(1 / (x² - 4)) dx?

Accepted Answer

1/2 * ln(|x - 2|) - 1/2 * ln(|x + 2|) + C

Answer

1/2 * ln(|x + 2|) + C

Answer

1/4 * ln(|x - 2|) - 1/4 * ln(|x + 2|) + C

Answer

ln(|x - 2|) - ln(|x + 2|) + C

Question 2

Which of the following integration techniques is used for rational functions?

Accepted Answer

Partial fraction decomposition

Answer

Integration by substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 3

Which of the following is the integration formula for logarithmic functions?

Accepted Answer

∫ ln(x) dx = x ln(x) - x + C

Answer

∫ ln(x) dx = (1/x) ln(x) + C

Answer

∫ ln(x) dx = x^2 ln(x) + C

Answer

∫ ln(x) dx = ln(x^2) + C

Question 4

Evaluate the integral of (**x² + 2x + 1**) **dx**.

Accepted Answer

⅓x³ + x² + x + C

Answer

½x³ + x² + x + C

Answer

¼x³ + x² + x + C

Answer

⅕x³ + x² + x + C

Question 5

Find the definite integral of **sin(x)** **dx** from **0** to **π/2**.

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 6

Which formula represents the technique of **integration by substitution**?

Accepted Answer

∫ f(g(x)) * g'(x) * dx

Answer

∫ f'(g(x)) * g(x) * dx

Answer

∫ g(f(x)) * f'(x) * dx

Answer

∫ f(x) * g'(x) * dx

Question 7

Which of the following is **NOT** a fundamental **rule of integration**?

Accepted Answer

The Product Rule

Answer

The Power Rule

Answer

The Sum Rule

Answer

The Quotient Rule

Question 8

Evaluate the **improper integral** ∫ (1 / **x**) **dx** from **1** to **∞**.

Accepted Answer

Diverges

Answer

0

Answer

1

Answer

∞

Question 9

Identify the integral that represents a **rational function integral**.

Accepted Answer

∫ (x² + 1) / (x - 1) dx

Answer

∫ sin(x) dx

Answer

∫ e^x dx

Answer

∫ √x dx

Question 10

Which of the following formulas represents integration by substitution?

Accepted Answer

∫f(u)du = F(u) + C

Answer

∫f(x)dx = F(x) + C

Answer

∫f'(u)du = F(u) + C

Answer

None of the provided choices are correct.

Question 11

Find the integral of ∫cos(x)sin(x)dx.

Accepted Answer

½sin²(x) + C

Answer

½cos²(x) + C

Answer

-(½)cos²(x) + C

Answer

None of the provided choices are correct.

Question 12

Evaluate the integral ∫ln(x)dx.

Accepted Answer

xln(x) - x + C

Answer

xln(x) + x + C

Answer

ln(x) + C

Answer

None of the provided choices are correct.

Question 13

Identify the appropriate substitution to solve the integral of ∫(x^2 + 1)^3 * x dx.

Accepted Answer

u = x^2 + 1

Answer

u = x

Answer

u = x^3

Answer

u = x^2

Question 14

Find the integral of ∫(sin(2x) * cos(x)) dx.

Accepted Answer

1/2 * sin(3x) + C

Answer

-1/2 * sin(3x) + C

Answer

1/2 * cos(3x) + C

Answer

-1/2 * cos(3x) + C

Question 15

Find the integral of ∫(e^(2x) / (1 + e^(2x))) dx.

Accepted Answer

ln(|1 + e^(2x)|) + C

Answer

-ln(|1 + e^(2x)|) + C

Answer

1 / (1 + e^(2x)) + C

Answer

-1 / (1 + e^(2x)) + C

Question 16

Which integral represents the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis?

Accepted Answer

2π∫[0,2](4 - x^2) dx

Answer

π∫[0,2](4 - x^2) dx

Answer

π∫[0,4](x^2 - 4) dx

Answer

2π∫[0,4](x^2 - 4) dx

Question 17

Which integration technique is appropriate for evaluating ∫(x^2 + 1)/(x + 1) dx?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Substitution

Question 18

Find the integral by substitution: ∫(x^2 + 1)^3 2x dx

Accepted Answer

(x^2 + 1)^4/2 + C

Answer

(x^2 + 1)^4 + C

Answer

(x^2 + 1)^2 + C

Answer

1/2(x^2 + 1)^4 + C

Question 19

Evaluate the definite integral: ∫[0, 1] (e^x + sin x) dx

Accepted Answer

e - cos 1

Answer

e + 1

Answer

2e - 1

Answer

1 - e

Question 20

Which technique is used to integrate the expression ∫(x * ln x) dx?

Accepted Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Partial fractions

Answer

U-substitution

Question 21

Find the integral by trigonometric substitution: ∫√(1 - x^2) dx

Accepted Answer

1/2 arcsin x + C

Answer

arcsin x + C

Answer

sin^-1 x + C

Answer

1/2 sin^-1 x + C

Question 22

Evaluate the integral: ∫(1 / (x^2 + 4)) dx

Accepted Answer

1/2 arctan(x/2) + C

Answer

arctan(x/2) + C

Answer

tan^-1(x/2) + C

Answer

1/2 tan^-1(x/2) + C

Question 23

Which of the following is the integral of e^(3x - 2)?

Accepted Answer

(1/3)e^(3x - 2) + C

Answer

3e^(3x - 2) + C

Answer

e^(3x - 2) / 3 + C

Answer

e^(3x - 2) * 3 + C

Question 24

Determine the integral of (x^2 + 3x - 5) dx.

Accepted Answer

x^3/3 + (3x^2)/2 - 5x + C

Answer

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

Answer

(x^3)/2 + (3x^2)/2 - 5x + C

Question 25

Which of the following integrals requires the use of trigonometric substitution?

Accepted Answer

∫√(16 - x^2) dx

Answer

∫e^x * sin(x) dx

Answer

∫x^2 * ln(x) dx

Answer

∫(x^2 + 1)^-1 dx

Question 26

Find the integral of ∫(cos^2(x) + sin^2(x)) dx.

Accepted Answer

x + C

Answer

sin^2(x) + cos^2(x) + C

Answer

-x + C

Answer

sin(x) + cos(x) + C

Question 27

What is the integral of 1/x dx?

Accepted Answer

ln|x| + C

Answer

x + C

Answer

1/x^2 + C

Answer

-1/x^2 + C

Question 28

Using integration by parts, find the integral of x*e^x dx.

Accepted Answer

x*e^x - e^x + C

Answer

x^2*e^x + C

Answer

e^x + C

Answer

x*e^x + C

Question 29

What is the integral of sec^2(x) dx?

Accepted Answer

tan(x) + C

Answer

sec(x) + C

Answer

-tan(x) + C

Answer

cot(x) + C

Question 30

What is the integral of 1/(1 + x^2) dx?

Accepted Answer

arctan(x) + C

Answer

arcsin(x) + C

Answer

ln(1 + x^2) + C

Answer

arccos(x) + C

Question 31

What is the integral of (x^2 + 1)/(x^3 + 3x) dx?

Accepted Answer

1/3 * ln|x^3 + 3x| + C

Answer

x^3 + 3x + C

Answer

1/3 * (x^3 + 3x) + C

Answer

ln|x^3 + 3x| + C

Question 32

Which of the following is a correct application of the substitution rule to solve the integral ∫x^2 * sqrt(x^3 + 1) dx?

Accepted Answer

u = x^3 + 1, du = 3x^2 dx

Answer

u = x^2, du = 2x dx

Answer

u = x^3, du = 3x^2 dx

Answer

u = sqrt(x^3 + 1), du = 3x^2 / (2*sqrt(x^3 + 1)) dx

Question 33

What is the integral of cos^2(x) dx?

Accepted Answer

x/2 + (sin(2x))/4 + C

Answer

sin^2(x) + C

Answer

cos(2x) + C

Answer

sin(x) + C

Question 34

What is the integral of e^(2x) dx?

Accepted Answer

1/2 * e^(2x) + C

Answer

2 * e^(2x) + C

Answer

e^x + C

Answer

e^(2x) + C

Question 35

What is the integral of  $\frac{1}{x}$  with respect to x?

Accepted Answer

$ln|x| + C$

Answer

$x + C$

Answer

$e^x + C$

Answer

$\frac{1}{x^2} + C$

Question 36

Which of the following is the result of integrating  $x^2 + 2x - 3$  with respect to x?

Accepted Answer

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

Answer

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

Answer

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

Answer

$2x + 2 + C$

Question 37

What is the integral of  $sin(x)$  with respect to x?

Accepted Answer

$-cos(x) + C$

Answer

$-sin(x) + C$

Answer

$tan(x) + C$

Answer

$cos(x) + C$

Question 38

Find the integral of  $e^{2x}$  with respect to x.

Accepted Answer

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

Answer

$2e^{2x} + C$

Answer

$e^{2x} + C$

Answer

$e^x + C$

Question 39

Using the substitution  $u = 2x + 1$, what is the integral of $\frac{1}{2x + 1}$  with respect to x?

Accepted Answer

$\frac{1}{2}ln|2x + 1| + C$

Answer

$ln|2x + 1| + C$

Answer

$\frac{1}{2}ln|x| + C$

Answer

$ln|x| + C$

Question 40

What is the integral of  $x cos(x)$  with respect to x using integration by parts?

Accepted Answer

$x sin(x) + cos(x) + C$

Answer

$-x cos(x) + sin(x) + C$

Answer

$x sin(x) - cos(x) + C$

Answer

$x cos(x) + sin(x) + C$

Question 41

Find the integral of $\frac{1}{\sqrt{1 - x^2}}$  with respect to x.

Accepted Answer

$arcsin(x) + C$

Answer

$arctan(x) + C$

Answer

$arccos(x) + C$

Answer

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

Question 42

Evaluate the definite integral $\int_0^1 (x^2 + 1) dx$.

Accepted Answer

$\frac{4}{3}$

Answer

$\frac{5}{3}$

Answer

$2$

Answer

$\frac{2}{3}$

Question 43

What is the area under the curve $f(x) = x^3$  from x = 0 to x = 2?

Accepted Answer

$4$

Answer

$1$

Answer

$2$

Answer

$8$

Question 44

Which of the following is NOT a correct statement about integration?

Accepted Answer

Integration is the process of finding the derivative of a function.

Answer

Integration is the opposite of differentiation.

Answer

Integration can be used to find the area under a curve.

Question 45

What is the integral of ∫(3x² + 2x - 1) dx?

Accepted Answer

x³ + x² - x + C

Answer

x³ + x² - 1 + C

Answer

6x + 2 + C

Answer

9x² + 4x + C

Question 46

Which of the following is the integral of ∫sin(2x) dx?

Accepted Answer

-1/2 cos(2x) + C

Answer

cos(2x) + C

Answer

1/2 cos(2x) + C

Answer

-cos(2x) + C

Question 47

Find the integral of ∫x e^x dx using integration by parts.

Accepted Answer

x e^x - e^x + C

Answer

x e^x + C

Answer

e^x + C

Question 48

What is the integral of ∫(1 / (x² + 1)) dx?

Accepted Answer

arctan(x) + C

Answer

1 / (x² + 1) + C

Answer

ln(x² + 1) + C

Answer

x / (x² + 1) + C

Question 49

Evaluate the integral ∫(x / (x² + 1)) dx using substitution.

Accepted Answer

1/2 ln(x² + 1) + C

Answer

ln(x² + 1) + C

Answer

x² + 1 + C

Question 50

What is the integral of ∫(1 / (x√(x² - 1))) dx?

Accepted Answer

arcsec(x) + C

Answer

ln(x² - 1) + C

Answer

arcsin(x) + C

Answer

arctan(x) + C

Question 51

If ∫f(x) dx = F(x) + C, then what is ∫f(2x) dx?

Accepted Answer

1/2 F(2x) + C

Answer

F(x) + C

Answer

2F(2x) + C

Answer

F(2x) + C

Question 52

Which of the following is the definite integral of ∫(x²) dx from 0 to 1?

Accepted Answer

1/3

Answer

1

Answer

2/3

Question 53

Evaluate the integral ∫(1 / (x - 1)) dx from 2 to 4.

Accepted Answer

ln(3)

Answer

ln(6)

Answer

ln(4)

Answer

ln(2)

Question 54

The integral of an odd function over a symmetric interval is always:

Accepted Answer

Zero

Answer

Positive

Answer

Negative