Chapter 5: TB. 2 (page 427)

Review of integration by substitution: Use u-substitution to find each of the following integrals.
a. $\int e^{3 x + 1} d x$ b. $\int x e^{x^{2} + 1} d x$
c. $\int \frac{\ln x}{x} d x$ d. $\int \frac{1}{x \ln x} d x$
e. $\int \tan x \sec^{2} x d x$ f. $\int \frac{1}{x^{2} + 1} d x$
g. $\int \sin x \sqrt{\cos x} d x$ h. $\int e^{x} \sin e^{x} d x$

Short Answer

Expert verified

Part (a) $\int e^{3 x + 1} d x = \frac{e^{3 x + 1}}{3} + C$

Part (b) $\int x e^{x^{2} + 1} d x = \frac{e^{x^{2} + 1}}{2} + C$

Part (c) $\int \frac{\ln x}{x} d x = \frac{{(\ln x)}^{2}}{2} + C$

Part (d) $\int \frac{1}{x \ln x} d x = \ln |\ln x| + C$

Part (e) $\int \tan x \sec^{2} x d x = \frac{\tan^{2} x}{2} + C$

Part (f) $\int \frac{1}{x^{2} + 1} d x = \tan^{- 1} x + C$

Part (g) $\int \sin x \sqrt{\cos x} d x = - \frac{2 \cos^{\frac{3}{2}} x}{3} + C$

Part (h) $\int e^{x} \sin e^{x} d x = - \cos (e^{x}) + C$

Step by step solution

Part (a) Step 1. Calculating the integral

Given integral, $\int e^{3 x + 1} d x$

Substituting, $3 x + 1 = u, 3 d x = d u$

$\int \frac{e^{u}}{3} d u = \frac{e^{u}}{3} + C = \frac{e^{3 x + 1}}{3} + C$

Part (b) Step 1. Calculating the integral

Given integral, $\int x e^{x^{2} + 1} d x$

Substituting, $x^{2} + 1 = u, 2 x d x = d u$

$\int \frac{e^{u}}{2} d u = \frac{e^{u}}{2} + C = \frac{e^{x^{2} + 1}}{2} + C$

Part (c) Step 1. Calculating the integral

Given integral, $\int \frac{\ln x}{x} d x$

Substituting, $\ln x = u, \frac{d x}{x} = d u$

$\int u d u = \frac{u^{2}}{2} + C = \frac{{(\ln x)}^{2}}{2} + C$

Part (d) Step 1. Calculating the integral

Given integral, $\int \frac{1}{x \ln x} d x$

Substituting, $\ln x = u, \frac{d x}{x} = d u$

$\int \frac{1}{u} d u = \ln |u| + C = \ln |\ln x| + C$

Part (e) Step 1. Calculating the integral

Given integral, $\int \tan x \sec^{2} x d x$

Substituting, $\tan x = u, s e c^{2} x d x = d u$

$\int u d u = \frac{u^{2}}{2} + C = \frac{\tan^{2} x}{2} + C$

Part (f) Step 1. Calculating the integral

Given integral, $\int \frac{1}{x^{2} + 1} d x$

$= \tan^{- 1} x + C$

Part (g) Step 1. Calculating the integral

Given integral, $\int \sin x \sqrt{\cos x} d x$

Substituting, $\cos x = u, - \sin x d x = d u$

$- \int \sqrt{u} d u = - \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C = - \frac{2 \cos^{\frac{3}{2}} x}{3} + C$

Part (h) Step 1. Calculating the integral

Given integral, $\int e^{x} \sin e^{x} d x$

Substituting, $e^{x} = u, e^{x} d x = d u$

$\int \sin u d u = - \cos u + C = - \cos (e^{x}) + C$

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Short Answer

Step by step solution

Part (a) Step 1. Calculating the integral

Part (b) Step 1. Calculating the integral

Part (c) Step 1. Calculating the integral

Part (d) Step 1. Calculating the integral

Part (e) Step 1. Calculating the integral

Part (f) Step 1. Calculating the integral

Part (g) Step 1. Calculating the integral

Part (h) Step 1. Calculating the integral

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Review of integration by substitution: Use u-substitution to find each of the following integrals.a. ∫e3x+1dxb. ∫xex2+1dxc. ∫lnxxdxd. ∫1xlnxdxe. ∫tanxsec2xdxf. ∫1x2+1dxg.∫sinxcosxdxh.∫exsinexdx

Short Answer

Step by step solution

Part (a) Step 1. Calculating the integral

Part (b) Step 1. Calculating the integral

Part (c) Step 1. Calculating the integral

Part (d) Step 1. Calculating the integral

Part (e) Step 1. Calculating the integral

Part (f) Step 1. Calculating the integral

Part (g) Step 1. Calculating the integral

Part (h) Step 1. Calculating the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Pure Maths

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Theoretical and Mathematical Physics

Discrete Mathematics

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