"Evaluate the following integral. ∫sec2xtan5xdx" solved and explained

Caelan

Answered question

2021-10-12

Evaluate the following integral.

\int \frac{\sec^{2} x}{\tan^{5} x} d x

tabuordy

Skilled2021-10-13Added 90 answers

Step 1
Evaluate,

\int \frac{\sec^{2} x}{\tan^{5} x} d x

Step 2
Simplification:
We have

\int \frac{\sec^{2} x}{\tan^{5} x} d x

...(1)
Substitute

\tan x = u =\Rightarrow \sec^{2} x d x = d u

in (1),

\int \frac{\sec^{2} x}{\tan^{5} x} d x = \int \frac{1}{u^{5}} d u

= \int u^{- 5} d u

= \frac{u^{- 5 + 1}}{- 5 + 1} + C

= \frac{u^{- 4}}{- 4} + C

= - \frac{1}{4 u^{4}} + C

= - \frac{1}{4 \tan^{4} x} + C

[Since

u = \tan x

]
Step 3
Hence

\int \frac{\sec^{2} x}{\tan^{5} x} d x = - \frac{1}{4 \tan^{4} x} + C

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