What is the integral of 4/(sqrt(e^t)) ?

The integral of 4/(sqrt(e^t)) is -8/(e^{t/2)}+C

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integral of 4/(sqrt(e^t))

\int \frac{4}{e ^{t}} d t

- \frac{8}{e ^{\frac{t}{2}}} + C

Solution steps

\int \frac{4}{e ^{t}} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{1}{e ^{t}} d t

Apply u-substitution

: \int \frac{1}{u u} d u

= 4 \cdot \int \frac{1}{u u} d u

Simplify

u u : u^{\frac{3}{2}}

= 4 \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

Apply the Power Rule

: - \frac{2}{u}

= 4 (- \frac{2}{u})

Substitute back

u = e^{t}

= 4 (- \frac{2}{e ^{t}})

Simplify

4 (- \frac{2}{e ^{t}}) : - \frac{8}{e ^{\frac{t}{2}}}

= - \frac{8}{e ^{\frac{t}{2}}}

Add a constant to the solution

= - \frac{8}{e ^{\frac{t}{2}}} + C

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What is the integral of 4/(sqrt(e^t)) ?
The integral of 4/(sqrt(e^t)) is -8/(e^{t/2)}+C