Use a table of integrals to find the indefinite integral ∫e3x(1+ex)3dx

Explained: "Use a table of integrals to find the..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

pro4ph5e4q2

Answered question

2021-12-04

Use a table of integrals to find the indefinite integral

\int \frac{e^{3 x}}{{(1 + e^{x})}^{3}} d x

Answer & Explanation

Supoilign1964

Beginner2021-12-05Added 19 answers

Step 1: Given that
Use a table of integrals to find the indefinite integral $\int \frac{e^{3 x}}{{(1 + e^{x})}^{3}} d x$
Step 2: Solve
We have,
$I = \int \frac{e^{3 x}}{{(1 + e^{x})}^{3}} d x$
Substitute,
$1 + e^{x} = t e^{x} * d x = d t$
Plugging all the values into the integral we obtain,
$I = \int \frac{e^{3 x}}{t^{3}} \times \frac{d t}{e^{x}}$
$= \int \frac{e^{2 x}}{t^{3}} d t = \int \frac{(t - 1)^{2}}{t^{3}} d t = \int \frac{t^{2} - 2 t + 1}{t^{3}} d t = \int \frac{d t}{t} - 2 \int \frac{d t}{t^{2}} + \int \frac{1}{t^{3}} d t$
$= \ln | t | - 2 (\frac{t^{- 2 + 1}}{- 2 + 1}) + (\frac{t^{- 3 + 1}}{- 3 + 1}) + C$
$= \ln | t | + \frac{2}{t} - \frac{1}{2 t^{2}} + C$
$= \ln | 1 + e^{x} | + \frac{2}{1 + e^{x}} - \frac{1}{2 {(1 + e^{x})}^{2}} + C$

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