Integrate the following functions w.r.t. x: x^9 ^2 (x^ 10 ) - Vidyadip

Question

Integrate the following functions w.r.t. x:
$x^9 \cdot \sec ^2\left(x^{10}\right)$

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Answer

Let $I=\int x^9 \cdot \sec ^2\left(x^{10}\right) d x$
Put $x^{10}=t \quad \therefore 10 x^9 d x=d t \quad \therefore x^9 d x=\frac{1}{10} d t$
$\therefore I=\int \sec ^2 t \cdot \frac{d t}{10}$
$=\frac{1}{10} \int \sec ^2 t d t$
$=\frac{1}{10} \tan t+c$
$=\frac{1}{10} \tan \left(x^{10}\right)+c .$

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