"Evaluate the following integrals. ∫sec4⁡xdx" was answered by our experts

a2linetagadaW

Answered question

2020-10-28

Evaluate the following integrals.

\int \sec^{4} x d x

Raheem Donnelly

Skilled2020-10-29Added 75 answers

It is given that, \int \sec^{4} x d x

We have to evaluate it.

We have, \int 6 \sec^{4} x d x

\Rightarrow \int 6 \sec^{4} x d x = 6 \int \sec^{4} x d x

\Rightarrow \int 6 \sec^{4} x d x = 6 \int \sec^{2} x \sec^{2} x d x

\Rightarrow \int 6 \sec^{4} x d x = 6 \int (1 + \tan^{2} x) \sec^{2} x d x,

Let \tan x = t

Differentiating equation, we get

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

Then, equation becomes

\Rightarrow \int 6 \sec^{4} x d x = 6 \int (1 + t^{2}) d t

\Rightarrow \int 6 \sec^{4} x d x = 6 [\int d t + \int t^{2} d t]

\Rightarrow \int 6 \sec^{4} x d x = 6 [t + \frac{t^{3}}{3}] + C

Substitute the value of t = 1 + \tan^{2} x in above equation,we get

\Rightarrow \int 6 \sec^{4} x d x = 6 [\tan x + \frac{\tan^{3} x}{3}] + C

\Rightarrow \int 6 \sec^{4} x d x = 6 \tan x + 2 \tan^{3} x + C

Hence, \int 6 \sec^{4} x d x = 6 \tan x + 2 \tan^{3} x + C

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