Question

Integrate the following:-

Answer 1

Given:

(1) ∫ (1+x)² / x(1+x²) dx

(2) ∫ (x^4+x^2+x) / 2(1+x²) dx

To find:

Integrate the following.

Solution:

From given, we have,

(1) ∫ (1+x)² / x(1+x²) dx

taking the partial fraction, we get,

= ∫ 1/x + 2/(x²+1) dx

= ∫ 1/x dx + ∫ 2/(x²+1) dx

= log x + 2tan^{-1}x + c

(2) ∫ (x^4+x^2+x) / 2(1+x²) dx

= 1/2 ∫ (x^4+x^2+x) / (1+x²) dx

using long division method, we get,

= 1/2 ∫ x² + 1/(1+x²) dx

= 1/2 [∫ x² dx + ∫ 1/(1+x²) dx]

= 1/2 [x³/3 + tan^{-1}x] + c

Answer 2

Answer:

Step-by-step explanation:

Hope you understood.