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$\int_{0}^{π / 2} \frac{\sin x}{1 + \cos^{2} x} dx$ is equal to

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a

π2

b

π4

c

3π2

d

π

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detailed solution

Correct option is B

Let I=∫0π/2 sin⁡x1+cos2⁡xdxput cos⁡x=t⇒−sin⁡xdx=dt⇒dx=−dtsin⁡xFor limit when x=0⇒t=cos⁡0=1 [∵t=cos⁡x]and when x=π2⇒t=cos⁡π2=0∴ I=∫10 sin⁡x1+t2⋅dt−sin⁡x.=−∫10 11+t2dt=∫01 11+t2dt=11tan−1⁡t110 ∵∫1a2+x2dx=1atan−1⁡xa=−0−π4=π4

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If $m \neq n, m, n \in N$ then the value of $\int_{0}^{2 π} \cos m x \cos n x d x$ is

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