Evaluate the following integrals: ^5x ^4xdx - Vidyadip

Question

Evaluate the following integrals:
$\int\tan^5\text{x }\sec^4\text{x}\text{dx}$

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Answer

$\int\tan^5\text{x }\sec^4\text{x}\text{dx}$
$=\int\tan^5\text{x }\sec^2\text{x}.\sec^2\text{x}\text{dx}$
$=\int\tan^5\text{x}(1+\tan^2\text{x})\sec^2\text{xdx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{xdx}=\text{dt}$
Now, $\int\tan^5\text{x}(1+\tan^2\text{x})\sec^2\text{xdx}$
$=\int\text{t}^5(1+\text{t}^2)\text{dt}$
$=\int(\text{t}^5+\text{t}^7)\text{dt}$
$=\frac{\text{t}^6}{6}+\frac{\text{t}^8}{8}+\text{C}$
$=\frac{\tan^6\text{x}}{6}+\frac{\tan^8\text{x}}{8}+\text{C}$

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