First slide

Evaluation of definite integrals

Question

_{$\int_{0}^{\infty} \frac{x}{(1 + x) (1 + x^{2})} d x =$}

Easy

A

_{$\frac{π}{4}$}
B

_$π$
C

_{$\frac{π}{2}$}
D

_{$2 π$}

Solution

_{$\int_{0}^{π / 2} \frac{T a n θ}{1 + T a n θ} d θ$} _{$\begin{matrix} x = \tan θ \\ d x = \sec^{2} θ \end{matrix}$}
_{$I = \int_{0}^{π / 2} \frac{S i n θ}{S i n θ + C o s θ} d θ$}

_{$I = \int_{0}^{π / 2} \frac{C o s θ}{C o s θ + S i n θ} d θ$}

_{$I = \int_{0}^{π / 2} d θ = \frac{π}{2}$}

_{$I = \frac{π}{4}$}

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