Master Integration Techniques: Substitution, Parts, and Partial Fractions

Question 1

Which of the following integrals can be solved using integration by parts?

Accepted Answer

∫xsin(x)dx

Answer

∫x^2e^xdx

Answer

∫tan(x)dx

Answer

∫sec^2(x)dx

Question 2

Which of the following substitutions can be used to simplify the integral ∫√(x^2 + 1)dx?

Accepted Answer

u = √(x^2 + 1)

Answer

u = x^2 + 1

Answer

u = x

Answer

u = 1/x

Question 3

Which of the following is NOT a technique of integration?

Accepted Answer

Exponentiation

Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Question 4

Which of the following integrals represents the area under the curve y = x^3 from x = 0 to x = 2?

Accepted Answer

∫(0 to 2)x^3dx

Answer

∫(2 to 0)x^3dx

Answer

∫(0 to 2)2x^3dx

Answer

∫(2 to 0)2x^3dx

Question 5

Evaluate the integral ∫(x / (x^2 + 1)) dx using partial fractions.

Accepted Answer

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

Answer

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

Answer

1/2 * arctan(x) + C

Answer

arctan(x) + C

Question 6

Evaluate the integral ∫∫∫(x + y + z) dV over the solid tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1

Accepted Answer

1/6

Answer

1/4

Answer

1/3

Answer

1/2

Question 7

Evaluate the line integral ∫c(x^2 + y^2) ds, where C is the parametrized curve r(t) = (cos(t), sin(t)), 0 ≤ t ≤ π/2

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 8

Question: Calculate the integral ∫x * e^(-x^2) dx. Explain why integration by parts is an appropriate substitution method for this integral.

Accepted Answer

Correct Answer: u = x, dv = e^(-x^2) dx

Answer

u = e^(-x^2), dv = x dx

Answer

u = x, dv = e^x dx

Answer

Integration by parts is not an appropriate choice for this integral.

Question 9

To evaluate the integral of f(x) = x^3 * ln(x), which integration technique should be used? Explain the steps involved in applying this technique.

Accepted Answer

Integration by parts, with u = ln(x) and dv = x^3 dx

Answer

Substitution, with u = x^3

Answer

Power rule of integration

Answer

The integral cannot be evaluated

Question 10

Evaluate the improper integral ∫[0, ∞) e^(-x) dx. Explain the concept of improper integrals and how the limit is used to determine if the integral converges or diverges.

Accepted Answer

The integral converges to 1.

Answer

The integral diverges.

Answer

The integral converges to e^(-∞).

Question 11

Evaluate the improper integral: ∫(1/x) dx from x = 1 to x = ∞

Accepted Answer

∞

Answer

0

Answer

-∞

Answer

1

Question 12

Which method is employed to integrate rational functions?

Accepted Answer

Partial fractions

Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 13

Integrate the exponential function: ∫e^(x + 1) dx

Accepted Answer

e^(x + 1) + C

Answer

e^x + C

Answer

xe^(x + 1) + C

Question 14

Which integral necessitates the use of the reverse chain rule?

Accepted Answer

∫sin(x^2) dx

Answer

∫x^3 dx

Answer

∫e^x dx

Answer

∫cos(x) dx

Question 15

Which of the following integrals requires the technique of integration by parts?

Accepted Answer

∫x^2 * sin(x) dx

Answer

∫(x^2 + 1) / (x + 1) dx

Answer

∫sin(x^2) dx

Answer

∫e^(2x) dx

Question 16

What is the appropriate substitution to solve the integral ∫x * √(x^2 + 1) dx?

Accepted Answer

u = x^2 + 1

Answer

u = √(x^2 + 1)

Answer

u = x

Answer

u = x^2

Question 17

Using integration by parts, the integral of ∫x * e^x dx is:

Accepted Answer

x * e^x - e^x + C

Answer

x * e^x + e^x + C

Answer

x * e^x + C

Answer

e^x + C

Question 18

What is the partial fraction decomposition of the rational function (2x + 1) / (x^2 - 1)?

Accepted Answer

1 / (x - 1) + 1 / (x + 1)

Answer

2 / (x - 1) - 1 / (x + 1)

Answer

1 / (x - 1) - 1 / (x + 1)

Question 19

The integral of ∫(x^2 + 2x) / (x^3 + 3x^2) dx is:

Accepted Answer

(1/3) * ln|x^3 + 3x^2| + C

Answer

ln|x + 3| + C

Answer

ln|x^3 + 3x^2| + C

Answer

(1/3) * ln|x + 3| + C

Question 20

The indefinite integral of ∫cos(3x) dx is:

Accepted Answer

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

Answer

sin(3x) + C

Answer

3 * sin(3x) + C

Answer

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

Question 21

Which of the following integrals involves a trigonometric substitution?

Accepted Answer

∫√(4 - x^2) dx

Answer

∫x * sin(x) dx

Answer

∫e^(2x) dx

Answer

∫(x^2 + 1) / (x + 1) dx

Question 22

The definite integral of ∫(x^2 + 1) dx from 0 to 1 is:

Accepted Answer

4/3

Answer

1

Answer

2/3

Answer

5/3

Question 23

The integral of ∫(x^3 + 2x^2 + x) / (x^2 + 1) dx is:

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 24

Evaluate the integral ∫ x sin(x²) dx.

Accepted Answer

-(1/2)cos(x²)+C

Answer

2x cos(x²)+C

Answer

cos(x²)+C

Answer

x² cos(x²)+C

Question 25

Evaluate the integral ∫ x² ln(x) dx.

Accepted Answer

(1/3)x³ln(x) - (1/9)x³ + C

Answer

x³ln(x) + C

Answer

x²ln(x) - (1/2)x² + C

Answer

(1/3)x³ln(x) + C

Question 26

Evaluate the integral ∫ (2x+1)/(x²+x-2) dx.

Accepted Answer

ln|x+2| + ln|x-1| + C

Answer

(2x+1)/(x²+x-2)² + C

Answer

ln|x²+x-2| + C

Answer

ln|x+1| + ln|x-2| + C

Question 27

Which of the following integrals requires integration by parts?

Accepted Answer

∫ x³e^(-x) dx

Answer

∫ sin³(x) cos(x) dx

Answer

∫ (x+1)/(x²-1) dx

Answer

∫ (x²+1)/(x³+x) dx

Question 28

What is the appropriate substitution to solve the integral ∫ sin(√x) / √x dx?

Accepted Answer

u = √x

Answer

u = sin(√x)

Answer

u = x²

Answer

u = x

Question 29

Evaluate the integral ∫ (x+1)/(x²-4) dx.

Accepted Answer

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

Answer

ln|x²-4| + C

Answer

ln|x-2| + ln|x+2| + C

Answer

(x+1)/(x²-4)² + C

Question 30

Which of the following is NOT a valid integration technique?

Accepted Answer

Integration by division

Answer

Substitution

Answer

Partial fractions

Answer

Integration by parts

Question 31

Evaluate the integral ∫ e^x sin(x) dx.

Accepted Answer

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

Answer

-e^x cos(x) + C

Answer

e^x sin(x) + C

Question 32

What is the integral of ∫ sec²(x) tan(x) dx?

Accepted Answer

(1/2)sec²(x) + C

Answer

sec(x) + C

Answer

-cos²(x) + C

Answer

tan(x) + C

Question 33

The integral ∫ 1/(x²-1) dx can be evaluated using:

Accepted Answer

Partial fractions

Answer

Trigonometric substitution

Answer

Integration by parts

Answer

Substitution