Get the solution to "Evaluate the integral. ∫tan4xsec6xdx"

Annette Arroyo

Answered question

2021-11-10

Evaluate the integral.

\int \tan^{4} x \sec^{6} x d x

un4t5o4v

Skilled2021-11-11Added 105 answers

Step 1
To evaluate the integral:

\int \tan^{4} x \sec^{6} x d x

Let substitute

t = \tan x

\frac{d t}{d x} = \sec^{2} x

d t = \sec^{2} x d x

Substituting the value,

\int \tan^{4} x \sec^{6} x d x = \int {(t)}^{4} \sec^{4} x d t

= \int t^{4} {(1 + \tan^{2} x)}^{2} d t

= \int t^{4} {(1 + t^{2})}^{2} d t

= \int t^{4} (1 + 2 t^{2} + t^{4}) d t

= \int (t^{8} + 2 t^{6} + t^{4}) d t

= \frac{t^{9}}{9} + \frac{2}{7} t^{7} + \frac{t^{5}}{5} + c

Step 2
Putting value back,

\int \tan^{4} x \sec^{6} x d x = \frac{1}{9} \tan^{9} x + \frac{2}{7} \tan^{7} x + \frac{1}{5} \tan^{5} x + c

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