What is the integral of 4/(sqrt(a^x)) ?

The integral of 4/(sqrt(a^x)) is -8/(a^{x/2)ln(a)}+C

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integral of 4/(sqrt(a^x))

\int \frac{4}{a ^{x}} d x

- \frac{8}{a ^{\frac{x}{2}} ln ( a )} + C

Solution steps

\int \frac{4}{a ^{x}} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{a ^{x}} d x

Apply u-substitution

: \int \frac{2}{u ^{2} ln ( a )} d u

= 4 \cdot \int \frac{2}{u ^{2} ln ( a )} d u

Take the constant out:

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

= 4 \cdot \frac{2}{ln ( a )} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 4 \cdot \frac{2}{ln ( a )} (- \frac{1}{u})

Substitute back

u = a^{x}

= 4 \cdot \frac{2}{ln ( a )} (- \frac{1}{a ^{x}})

Simplify

4 \cdot \frac{2}{ln ( a )} (- \frac{1}{a ^{x}}) : - \frac{8}{a ^{\frac{x}{2}} ln ( a )}

= - \frac{8}{a ^{\frac{x}{2}} ln ( a )}

Add a constant to the solution

= - \frac{8}{a ^{\frac{x}{2}} ln ( a )} + C

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