Answered: "Integral Calculus: Evaluate the following examples of power..."

melodykap

Answered question

2021-11-07

Integral Calculus: Evaluate the following examples of power formula, exponential function, and logarithmic function.

\int e^{x} {(4 - e^{x})}^{\frac{3}{2}} d x

Nichole Watt

Skilled2021-11-08Added 100 answers

Step 1
Consider the given integral,

I = \int e^{x} {(4 - e^{x})}^{\frac{3}{2}} d x

use the substitution,

4 - e^{x} = t

- e^{x} d x = d t

so, the integral becomes,

I = - \int t^{\frac{3}{2}} d t

Step 2
now, using the integral formula,

I = - (\frac{t^{\frac{3}{2} + 1}}{\frac{3}{2} + 1}) + C

= - \frac{2}{5} t^{\frac{5}{2}} + C

now, undo the substitution,

I = - \frac{2}{5} {(4 - e^{x})}^{52} + C

hence the value of the integral is

- \frac{2}{5} {(4 - e^{x})}^{52} + C

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