Name: Method of Partial Fractions
Uploaded: 2019-07-19
Duration: 5 min 36 s
Description: Watch this video to understand how proper rational functions are integrated when the denominator consists of a linear or a quadratic expression.

Question

Evaluate:

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

Accepted Answer

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

Now putting, $\frac{x^{2} + x + 1}{(x + 2) (x^{2} + 1)} = \frac{A}{(x + 2)} + \frac{B x + C}{(x^{2} + 1)}$

$\Rightarrow A (x^{2} + 1) + (B x + C) (x + 2) = x^{2} + x + 1$

Putting $x + 2 = 0 \Rightarrow x = - 2$

$\Rightarrow A (5) + 0 = 4 - 2 + 1$

$\Rightarrow A = \frac{3}{5}$

Comparing constants, we get

$A + 2 C = 1$

$\Rightarrow 2 C = 1 - \frac{3}{5}$

$\Rightarrow C = \frac{1}{5}$

Comparing the coefficients of term $x$ , we get

$2 B + C = 1$

$\Rightarrow 2 B = 1 - \frac{1}{5}$

$\Rightarrow B = \frac{2}{5}$

$I = \int \frac{x^{2} + x + 1}{(x + 2) (x^{2} + 1)} d x = \int \frac{A}{(x + 2)} d x + \int \frac{B x + C}{(x^{2} + 1)} d x$

$= \frac{3}{5} \times \int \frac{1}{(x + 2)} d x + \frac{1}{5} \times \int \frac{2 x + 1}{(x^{2} + 1)} d x$

$= \frac{3}{5} \log | x + 2 | + \frac{1}{5} I_{1} + C_{1}$

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

$= \log |x^{2} + 1| + \tan^{- 1} x + C_{2}$ $[∵ \int \frac{f^{'} (x)}{f (x)} d x = \log |f (x)|]$

$∴ \int \frac{x^{2} + x + 1}{(x + 2) (x^{2} + 1)} d x = \frac{3}{5} \log | x + 2 | + \frac{1}{5} \log |x^{2} + 1| + \frac{1}{5} \tan^{- 1} x + C$

Evaluate:

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

Important Questions on Integration Using Partial Fractions

Learn and Prepare for any exam you want !

Evaluate: ∫x2+x+1(x+2)(x2+1)dx

Important Questions on Integration Using Partial Fractions

Learn and Prepare for any exam you want !

Evaluate:

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