Resolved: "Evaluate the indefinite integral. ∫sec2(7x)tan7(7x)dx"

Khadija Wells

Answered question

2021-10-21

Evaluate the indefinite integral.

\int \sec^{2} (7 x) \tan^{7} (7 x) d x

Usamah Prosser

Skilled2021-10-22Added 86 answers

Step 1
It is required to evaluate the indefinite integral:

\int \sec^{2} (7 x) \tan^{7} (7 x) d x

Step 2
let

\tan (7 x) = u

differentiating both sides with respect to x

7 \sec^{2} (7 x) = \frac{d u}{d x} \Rightarrow \sec^{2} (7 x) d x = \frac{d u}{7}

Step 3
Now put these values in the given integral:

\int \sec^{2} (7 x) \tan^{7} (7 x) d x = \int \frac{u^{7} d u}{7}

= \frac{1}{7} \int u^{7} d u

= \frac{1}{7} \frac{u^{8}}{8} + c

= \frac{1}{56} (u^{8}) + c

, c is a constant
Step 4
Now, substitute the value of u back:

\int \sec^{2} (7 x) \tan^{7} (7 x) d x = \frac{1}{56} (u^{8}) + c

= \frac{1}{56} (\tan^{8} (7 x)) + c

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