The value of [a^x+a^ -x ]^2 d x is: - Vidyadip

MCQ

The value of$\int\left[a^x+a^{-x}\right]^2 d x$ is:

A
$\frac{a^x}{2}+\frac{a^{-x}}{2}+2 x+C$
B
$\frac{a^{2 x}}{2}+\frac{a^{-2 x}}{2}+2 x+C$
✓
$\frac{a^{2 x}}{2 \log a}-\frac{a^{-2 x}}{2 \log a}+2 x+C$
D
$2 \frac{a^x}{\log a}-\frac{a^{-x}}{2 \log a}+2 x+C$

✓

Answer

Correct option: C.

$\frac{a^{2 x}}{2 \log a}-\frac{a^{-2 x}}{2 \log a}+2 x+C$

(C) $\frac{a^{2 x}}{2 \log a}-\frac{a^{-2 x}}{2 \log a}+2 x+C$
$\begin{aligned} I & =\int\left[a^x+a^{-x}\right]^2 d x \\ & =\int\left(a^{2 x}+a^{-2 x}+2 a^x a^{-x}\right) d x \\ & =\int\left(a^{2 x}+a^{-2 x}+2\right) d x \\ & =\frac{a^{2 x}}{2 \log a}+\frac{a^{-2 x}}{-2 \log a}+2 x+C \\ & =\frac{a^{2 x}}{2 \log a}-\frac{a^{-2 x}}{2 \log a}+2 x+C\end{aligned}$

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