Mathematics

Evaluate the following integrals:
$$\displaystyle \int { \sec ^{ 6 }{ x } \tan ^{  }{ x }  } dx$$


SOLUTION
$$\displaystyle\int{{\sec}^{6}{x}\tan{x}dx}$$

$$=\displaystyle\int{{\sec}^{5}{x}\sec{x}\tan{x}dx}$$

Let $$t=\sec{x}\Rightarrow\,dt=\sec{x}\tan{x}dx$$

$$=\displaystyle\int{{t}^{5}dt}$$

We know that $$\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$$

$$=\dfrac{1}{6}{t}^{6}+c$$

$$=\dfrac{1}{6}{\sec}^{6}{x}+c$$ where $$t=\sec{x}$$

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Subjective Medium Published on 17th 09, 2020
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