First slide

Introduction to integration

Question

$\int \sinh^{- 1} (\frac{x}{4}) d x =$

Easy

A

$x S i n h^{- 1} (\frac{x}{4}) - \sqrt{x^{2} + 16} + c$
B

$x S i n h^{- 1} (\frac{x}{2}) + \sqrt{x^{2} + 16} + c$
C

$x S i n h^{- 1} (\frac{x}{2}) - \frac{1}{2} \sqrt{x^{2} + 16} + c$
D

$x S i n h^{- 1} (\frac{x}{2}) - x \sqrt{x^{2} + 16} + c$

Solution

$\int \sinh^{- 1} (\frac{x}{4}) d x$ = $\sinh^{- 1} (\frac{x}{4}) x - \int \frac{x}{4} \frac{1}{\sqrt{1 + \frac{x^{2}}{16}}} d x$
$= x \sinh^{- 1} (\frac{x}{4}) - \frac{2}{4} \int \frac{2 x}{\sqrt{16 + x^{2}}} d x$

$= x \sinh^{- 1} \frac{x}{4} - \frac{1}{2} \times 2 \sqrt{16 + x^{2}} + c$

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