Chapter 1

Question 1

What is the formula for integration by substitution?

Answer 1

If u = g(x), then ∫ f(g(x)g’(x))dx = ∫ f(u)du

This technique simplifies integration by changing variables.

Question 2

What is the first step in the integration by substitution process?

Answer 2

Determine u = g(x)

This involves identifying a suitable function to substitute.

Question 3

How do you express dx in terms of du during substitution?

Answer 3

dx = (1/g’(x))du

This is derived from the relationship between u and x.

Question 4

What is the result of the integral ∫(x² + 2x + 3)6(x + 1)dx after substitution?

Answer 4

1/14(x² + 2x + 3)⁷ + C

C is the constant of integration.

Question 5

In the example ∫x⁻²(1 + 2/x)⁵dx, what substitution is made?

Answer 5

Let u = 1/x + 2

This simplifies the integral for easier evaluation.

Question 6

What is the integral of sin²(x)e^(cos²(x))dx using substitution?

Answer 6

-1/2 e^(cos²(x)) + C

The solution involves integrating with respect to u.

Question 7

What is the integral of 1/u du?

Answer 7

ln|u| + C

This is a standard result in integration involving logarithmic functions.

Question 8

What is the integral of e^u du?

Answer 8

e^u + C

This is a basic property of exponential functions.

Question 9

What is the rule for integration by parts?

Answer 9

∫u dv = uv - ∫v du

This rule corresponds to the product rule of differentiation.

Question 10

What is tabular integration?

Answer 10

A method used for repeated integration by parts

It simplifies the process when many repetitions are needed.

Question 11

What is the definition of improper integrals?

Answer 11

Integrals where the interval of integration is infinite or includes a singularity

This includes Type I and Type II improper integrals.

Question 12

What are the two types of improper integrals?

Answer 12

• Type I: Infinite Limits of Integration
• Type II: Discontinuous Integrand

These types help classify the behavior of the integral.

Question 13

What does it mean for an improper integral to converge?

Answer 13

The limit used to define the integral exists and is finite

Conversely, if the limit does not exist, it diverges.

Question 14

What is the improper integral ∫[1, ∞] (1/x^p)dx converges for what values of p?

Answer 14

p > 1

This determines the conditions under which the integral converges.

Question 15

In integration involving trigonometric functions, what is a common substitution for sin(x)?

Answer 15

u = cos(x)

This substitution simplifies integrals involving sine and cosine.

Question 16

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

Answer 16

sec²(x) + C

This is a standard integral in calculus involving trigonometric functions.

Question 17

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

Answer 17

-ln|cos(x)| + C

This can also be expressed as ln|sec(x)| + C.

Question 18

What is the method for evaluating improper integrals?

Answer 18

Calculate the limit of the integral as it approaches the point of discontinuity or infinity

This is essential for determining convergence.

Question 19

What is the formula for integration by substitution?

Answer 19

∫ f(g(x))g’(x)dx = ∫ f(u)du

Question 20

Ehherh

Answer 20

((x^2 + 2x + 3)^7)/7 + C

Question 22

What is the method of Integration by Substitution?

Answer 22

Integration by Substitution involves letting u = g(x) so that du = g′(x) dx, which transforms an integral ∫ f(g(x)) g′(x) dx into ∫ f(u) du.

Question 23

What is the formula for Integration by Parts?

Answer 23

Integration by Parts is given by ∫ u dv = uv − ∫ v du, derived from the product rule for differentiation.

Question 24

How are rational functions integrated using Partial Fractions?

Answer 24

The method involves decomposing a rational function into a sum of simpler fractions (with linear or quadratic denominators) that can be integrated individually.

Question 25

What are Trigonometric Techniques of Integration?

Answer 25

These techniques use trigonometric identities and substitutions to simplify integrals involving trigonometric functions into more manageable forms.

Question 26

What are Improper Integrals and how are they evaluated?

Answer 26

Improper Integrals involve infinite limits or discontinuous integrands and are evaluated by replacing the problematic limit with a variable and taking its limit (e.g. ∫ₐ∞ f(x) dx = lim₍b→∞₎ ∫ₐᵇ f(x) dx).

Question 27

Solve ∫ dx/(x² + 2x + 3) using substitution.

Answer 27

Complete the square: x² + 2x + 3 = (x+1)² + 2. Let u = x + 1 so that the integral becomes ∫ du/(u² + 2), which evaluates to (1/√2) arctan(u/√2) + C, i.e. (1/√2) arctan((x+1)/√2) + C.

Question 28

Evaluate ∫ (5x − 23)/(6x² − 11x − 7) dx using partial fractions.

Answer 28

Factor the denominator as (2x + 1)(3x − 7). Write (5x − 23)/(6x² − 11x − 7) = A/(2x + 1) + B/(3x − 7). Solving gives A = 3 and B = −2. Thus, the integral is (3/2) ln|2x + 1| − (2/3) ln|3x − 7| + C.

Question 29

Evaluate the definite integral ∫₋π/₃⁰ cos x · sin³ x dx.

Answer 29

Let u = sin x (so du = cos x dx). Changing limits: when x = −π/3, u = −√3/2; when x = 0, u = 0. The integral becomes ∫₋√3/₂⁰ u³ du = [u⁴/4]₋√3/₂⁰ = −(9/64), so the value is −9/64.

Question 30

For which values of p does the improper integral ∫₁∞ 1/xᵖ dx converge?

Answer 30

It converges if and only if p > 1; for p ≤ 1 the integral diverges.

Question 31

Does the improper integral ∫₁∞ 1/x dx converge?

Answer 31

No – since p = 1 the integral diverges (it grows logarithmically without bound).

Question 32

Is the area under y = (ln x)/x² from x = 1 to ∞ finite, and if so, what is it?

Answer 32

Yes, it is finite. Using integration by parts one finds ∫₁∞ (ln x)/x² dx = 1.

Question 33

What is Tabular Integration and when is it used?

Answer 33

Tabular Integration is a streamlined method for performing repeated integration by parts, especially when one function differentiates repeatedly to zero. It organizes derivatives and integrals in a table to simplify the process.