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Question: If I = \(\int_{- 1}^{1}\left( \frac{x^{2} + sinx}{1 + x^{2}} \right)\) dx then...

If I = ∫−11(x2+sinx1+x2)\int_{- 1}^{1}\left( \frac{x^{2} + sinx}{1 + x^{2}} \right) dx then

A

0

B

2

C

π/2

D

2 – π/2

Answer

2 – π/2

Explanation

Solution

I = ∫−11x21+x2dx+0(becausesin⁡x1+x2isodd)\int_{- 1}^{1}{\frac{x^{2}}{1 + x^{2}}dx + 0(because\frac{\sin x}{1 + x^{2}}}isodd)

I = 2∫01x2+1−1x2+12\int_{0}^{1}\frac{x^{2} + 1 - 1}{x^{2} + 1} d x = 2∫01dx−2∫01dx1+x2\int_{0}^{1}{dx} - 2\int_{0}^{1}\frac{dx}{1 + x^{2}}

= 2 – 2 tan–1 x ∣01 \left| {}_{0}^{1} \right.\ = 2 - π2\frac{\pi}{2}