Integration Techniques: Master Integration by Parts, Trig Substitution, and Partial Fractions

Question 1

Which of the following is NOT a valid integration technique?

Accepted Answer

Chain rule

Answer

u-substitution

Answer

Integration by parts

Answer

Partial fractions

Question 2

Find the integral: $$int \sqrt{x^2 + 1} dx$$

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 3

Which of the following integrals can be evaluated using the method of trigonometric substitution?

Accepted Answer

∫ cos(x) dx

Answer

∫ x^2 + 1 dx

Answer

∫ e^x dx

Question 4

Which of these techniques is best suited to find the integral of (x^3 + 2x^2 - 1)/(x - 1)?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Question 5

Which of the following is an appropriate substitution for the integral ∫(1 + tan(x))^3 sec^2(x) dx?

Accepted Answer

u = tan(x)

Answer

u = sec(x)

Answer

u = cos(x)

Answer

u = sin(x)

Question 6

Find the indefinite integral of ∫(x^3 + 2x^2 - 5x + 1) dx.

Accepted Answer

(1/4)x^4 + (2/3)x^3 - (5/2)x^2 + x + C

Answer

(1/4)x^4 - (2/3)x^3 + (5/2)x^2 + x + C

Answer

(1/4)x^4 + (2/3)x^3 + (5/2)x^2 - x + C

Answer

(1/4)x^4 - (2/3)x^3 - (5/2)x^2 - x + C

Question 7

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

Accepted Answer

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

Answer

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

Answer

tan^-1(x + 1) + C

Answer

tan^-1|x + 1| + C

Question 8

Which of the following is a necessary condition for a function to be integrable?

Accepted Answer

It must be continuous on the interval of integration.

Answer

It must be differentiable on the interval of integration.

Answer

Its graph must have no sharp corners.

Answer

Its antiderivative must exist.

Question 9

Evaluate the integral ∫(e^x + 1) / (e^x - 1) dx

Accepted Answer

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

Answer

ln(e^x - 1) + C

Answer

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

Answer

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

Question 10

Which of the following is an incorrect integration technique often used for integration?

Accepted Answer

Integration by improper rationalization

Answer

Integration by parts

Answer

Integration by trigonometric substitution

Answer

Integration by partial fractions

Question 11

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

Accepted Answer

∫ x sin(x) dx

Answer

∫ e^x dx

Answer

∫ x³ dx

Answer

∫ 1 / x dx

Question 12

The integral of (x + 1) / (x² - x + 1) dx can be solved using:

Accepted Answer

Partial fractions

Answer

Trigonometric substitution

Answer

Logarithmic substitution

Answer

Integration by parts

Question 13

The integral of sin(x) cos(x) dx is:

Accepted Answer

1/2 sin(2x) + C

Answer

-1/2 cos(2x) + C

Answer

1/2 cos(2x) + C

Answer

-1/2 sin(2x) + C

Question 14

Which of the following is an indefinite integral?

Accepted Answer

∫ x dx + C

Answer

∫ x dx

Answer

x dx

Answer

x dx + 1

Question 15

Which integration technique is most appropriate for ∫(x^2)cos(x) dx?

Accepted Answer

Integration by Parts

Answer

Partial Fractions

Answer

Trigonometric Substitution

Question 16

Evaluate ∫tan(x) dx:

Accepted Answer

ln|sec(x)| + C

Answer

cos(x) + C

Answer

-1/cos(x) + C

Answer

sin(x) + C

Question 17

Decompose 1/(x^2 - 4) into partial fractions:

Accepted Answer

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

Answer

(x - 2)/(x^2 - 4)

Answer

1/(x - 2)^2

Answer

1/x - 1/(x + 2)

Question 18

Find ∫∫cos(y)sin(x) dydx over the region R bounded by y = 0, y = π/2, x = 0, and x = π:

Accepted Answer

1

Answer

2

Answer

0

Answer

-1

Question 19

What is the integral of ∫x^n dx?

Accepted Answer

x^(n+1)/(n+1) + C

Answer

nx^n + C

Answer

x^n + C

Answer

1/n x^n + C

Question 20

Which integral can be solved using integration by substitution?

Accepted Answer

∫(x^2 + 1)^(3/2) x dx

Answer

∫cos(x) dx

Answer

∫e^x dx

Answer

∫1/x dx

Question 21

Evaluate ∫∫(x + y) dA over the triangular region with vertices (0, 0), (2, 0), and (0, 4):

Accepted Answer

16

Answer

8

Answer

24

Question 22

Which integration technique is most suitable for ∫√(x^2 + 1) dx?

Accepted Answer

Trigonometric Substitution

Answer

U-substitution

Answer

Integration by Parts

Answer

Partial Fractions

Question 23

What is the appropriate integration technique to use for the integral ∫x sin(x) dx?

Accepted Answer

Integration by parts

Answer

U-substitution

Answer

Partial fractions

Answer

Trigonometric substitution

Question 24

What is the appropriate trigonometric substitution for the integral ∫√(9-x²) dx?

Accepted Answer

x = 3sin(θ)

Answer

x = 3sec(θ)

Answer

x = 3cos(θ)

Answer

x = 3tan(θ)

Question 25

Which of the following is a correct partial fraction decomposition of the rational function (2x+1)/(x²-1)?

Accepted Answer

A/(x-1) + B/(x+1)

Answer

(2x+1)/(x²-1)

Answer

A/(x-1) + B/(x+1)²

Answer

A/(x-1)² + B/(x+1)²

Question 26

Which of the following integrals requires both integration by parts and a trigonometric substitution?

Accepted Answer

∫x²cos(x) dx

Answer

∫√(1-x²) dx

Answer

∫(1/x²) dx

Answer

∫(1/(x²+1)) dx