Name: Integration by Substitution: Illustrations
Uploaded: 2019-07-19
Duration: 21 min 35 s
Description: With the basics of integration, it's time to learn about more complicated integration techniques! Let's learn the method of integration by substitution rule.

Question

Evaluate the following integral:

\int \frac{x^{2} + x + 1}{{(x + 1)}^{2} (x + 2)} d x

Accepted Answer

Given $\int \frac{x^{2} + x + 1}{{(x + 1)}^{2} (x + 2)} d x$

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

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

Let us solve these two integrals separately,

$\int \frac{x^{2}}{{(x + 1)}^{2} (x + 2)} d x$

Perform partial fraction decomposition:

$\frac{x^{2}}{{(x + 1)}^{2} (x + 2)} = \frac{a}{x + 1} + \frac{b}{{(x + 1)}^{2}} + \frac{c}{x + 2}$

$\Rightarrow x^{2} = a (x + 1) (x + 2) + b (x + 2) + c {(x + 1)}^{2}$

$\Rightarrow x^{2} = x^{2} (a + c) + x (3 a + b + 2 c) + 2 a + 2 b + c$

Compare the coefficients on both sides

$a + c = 1; 3 a + b + 2 c = 0; 2 a + 2 b + c = 0$

$\Rightarrow a = 1 - c$

$∴ 3 - 3 c + b + 2 c = 0 \Rightarrow c = b + 3$

$\Rightarrow 1 - 2 (b + 3) + 2 b + b + 3 = 0 \Rightarrow b = 1$

$\Rightarrow c = 4, a = - 3$

$∴ \frac{x^{2}}{{(x + 1)}^{2} (x + 2)} = - \frac{3}{x + 1} + \frac{1}{{(x + 1)}^{2}} + \frac{4}{x + 2}$

$\Rightarrow \int \frac{x^{2}}{{(x + 1)}^{2} (x + 2)} d x = - 3 \int \frac{1}{x + 1} d x + \int \frac{1}{{(x + 1)}^{2}} d x + 4 \int \frac{1}{x + 2} d x$

$= - 3 \ln |x - 1| - \frac{1}{x + 1} + 4 \ln |x + 2|$

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

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

Evaluate the following integral: $\int \frac{x^{2} + x + 1}{{(x + 1)}^{2} (x + 2)} d x$

Important Questions on Indefinite Integrals

Learn and Prepare for any exam you want !

Evaluate the following integral: ∫x2+x+1x+12x+2dx

Important Questions on Indefinite Integrals

Learn and Prepare for any exam you want !

Evaluate the following integral: $\int \frac{x^{2} + x + 1}{{(x + 1)}^{2} (x + 2)} d x$