First slide

Evaluation of definite integrals

Question

_{$\int_{0}^{1} \frac{x^{5}}{1 + x^{4}} d x =$}

Easy

A

_{$\frac{2 - π}{4}$}
B

_{$\frac{2 + π}{4}$}
C

_{$\frac{4 + π}{8}$}
D

_{$\frac{4 - π}{8}$}

Solution

_{$\int_{0}^{1} \frac{x^{5}}{x^{4} + 1} d x$} _{$\begin{matrix} x^{2} = t, \\ 2 x d x = d t \end{matrix}$}
_{$= \int_{0}^{1} \frac{t^{2}}{t^{2} + 1} \frac{d t}{2}$}

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

_{$= \frac{1}{2} [{(t)}_{0}^{1} - {(T a n^{- 1} t)}_{0}^{1}]$}

_{$= \frac{1}{2} (1 - \frac{π}{4}) = \frac{1}{2} - \frac{π}{8}$}

_{$= \frac{4 - π}{8}$}

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