Chapter 5: TB. 2 (page 440)

Integration: Calculate the following integrals.
a. $\int \frac{1}{{(x - 1)}^{2}} d x$ b. $\int \frac{x - 1}{x^{2} - 1} d x$
c. $\int \frac{1}{x^{2} + 1} d x$ d. $\int \frac{x}{x^{2} + 1} d x$
e. $\int \frac{x^{2} + 1}{x^{3}} d x$ f. $\int \frac{x + 1}{x^{2} + 1} d x$

Short Answer

Expert verified

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

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

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

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

Part (e) $\int \frac{x^{2} + 1}{x^{3}} d x = \ln |x| - \frac{1}{2 x^{2}} + C$

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

Step by step solution

Part (a) Step 1. Calculating the integral

Given integral, $\int \frac{1}{{(x - 1)}^{2}} d x$

$= \frac{{(x - 1)}^{- 2 + 1}}{(- 2 + 1)} = - \frac{1}{x - 1} + C$

Part (b) Step 1. Calculating the integral

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

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

Part (c) Step 1. Calculating the integral

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

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

Part (d) Step 1. Calculating the integral

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

$= \int \frac{2 x}{2 (x^{2} + 1)} d x = \frac{\ln | x^{2} + 1 |}{2} + C$

Part (e) Step 1. Calculating the integral

Given integral, $\int \frac{x^{2} + 1}{x^{3}} d x$

$= \int \frac{x^{2}}{x^{3}} + \frac{1}{x^{3}} d x = \int \frac{1}{x} + \frac{1}{x^{3}} d x = \ln |x| + \frac{x^{- 3 + 1}}{(- 3 + 1)} + C = \ln |x| - \frac{1}{2 x^{2}} + C$

Part (f) Step 1. Calculating the integral

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

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

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Integration: Calculate the following integrals.
a. $\int \frac{1}{{(x - 1)}^{2}} d x$ b. $\int \frac{x - 1}{x^{2} - 1} d x$
c. $\int \frac{1}{x^{2} + 1} d x$ d. $\int \frac{x}{x^{2} + 1} d x$
e. $\int \frac{x^{2} + 1}{x^{3}} d x$ f. $\int \frac{x + 1}{x^{2} + 1} d x$

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

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Integration: Calculate the following integrals.a.∫1x-12dxb.∫x-1x2-1dxc.∫1x2+1dxd.∫xx2+1dxe.∫x2+1x3dxf.∫x+1x2+1dx

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

One App. One Place for Learning.

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Integration: Calculate the following integrals.
a. $\int \frac{1}{{(x - 1)}^{2}} d x$ b. $\int \frac{x - 1}{x^{2} - 1} d x$
c. $\int \frac{1}{x^{2} + 1} d x$ d. $\int \frac{x}{x^{2} + 1} d x$
e. $\int \frac{x^{2} + 1}{x^{3}} d x$ f. $\int \frac{x + 1}{x^{2} + 1} d x$