First slide

Evaluation of definite integrals

Question

$\int_{0}^{1} \frac{x}{{(1 - x)}^{5 / 4}} d x =$

Easy

A

$\frac{16}{3}$
B

$- \frac{16}{3}$
C

$\frac{3}{16}$
D

$- \frac{3}{16}$

Solution

$\int_{0}^{1} \frac{x}{{(1 - x)}^{5 / 4}} d x =$
$p u t 1 - x = t \Rightarrow d x = - d t$
= $\int_{1}^{0} \frac{1 - t}{t^{5 / 4}} (- d t) = \int_{0}^{1} \frac{1 - t}{t^{5 / 4}} (d t) \int_{0}^{1} (t^{- 5 / 4} - t^{- 1 / 4}) d t p u t 1 - x = t \Rightarrow d x = - d t$
$= \frac{- 16}{3}$

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