Explained: "Evaluate the integral. ∫exsec2(ex−7)dx"

Rena Giron

Answered question

2021-11-08

Evaluate the integral.

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

Luis Sullivan

Beginner2021-11-09Added 11 answers

Step 1
Given:

I = \int e^{x} \sec^{2} (e^{x} - 7) d x

for evaluating given integral, we substitute

e^{x} - 7 = t

...(1)
now, differentiating equation (1) with respect to x

\frac{d}{d x} (e^{x} - 7) = \frac{d}{d x} (t)

\frac{d}{d x} (e^{x}) - \frac{d}{d x} (7) = \frac{d t}{d x} (∵ \frac{d}{d x} (e^{x}) = e^{x})

e^{x} - 0 = \frac{d t}{d x}

e^{x} d x = d t

Step 2
now, replacing

e^{x} d x

with dt,

(e^{x} - 7)

with t in given integral
so,

\int e^{x} \sec^{2} (e^{x} - 7) d x = \int \sec^{2} t d t (∵ \int \sec^{2} x d x = \tan x + c)

= \tan t + c

...(2)
now substituting

t = e^{x} - 7

in equation (2)

\int e^{x} \sec^{2} (e^{x} - 7) d x = \tan (e^{x} - 7) + c

hence, given integral is equal to

\tan (e^{x} - 7) + c

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?