Resolved: "Evaluate the following integrals. ∫3x2+2x+3x4+2x2+1dx"

Jaya Legge

Answered question

2021-10-14

Evaluate the following integrals.

\int \frac{3 x^{2} + 2 x + 3}{x^{4} + 2 x^{2} + 1} d x

rogreenhoxa8

Skilled2021-10-15Added 109 answers

Step 1
Given integral is

\int \frac{3 x^{2} + 2 x + 3}{x^{4} + 2 x^{2} + 1} d x

.
Step 2
Evaluate the integral as follows.

\int \frac{3 x^{2} + 2 x + 3}{x^{4} + 2 x^{2} + 1} d x = \int \frac{3 x^{2}}{x^{4} + 2 x^{2} + 1} d x + \int \frac{2 x}{x^{4} + 2 x^{2} + 1} d x + \int \frac{3}{x^{4} + 2 x^{2} + 1} d x

= 3 (\frac{1}{2} \tan^{- 1} (x) - \frac{x}{2 (1 + x^{2})}) - \frac{1}{x^{2} + 1} + \frac{3}{2} (\frac{1}{2} \tan^{- 1} (x) + \frac{x}{2 (1 + x^{2})}) + C

= - \frac{3 x}{2 (x^{2} + 1)} + 3 \tan^{- 1} x + \frac{3 x}{2 (x^{2} + 1)} - \frac{1}{x^{2} + 1} + C

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