What is the integral of 36e^{6x}sqrt(5+e^{6x)} ?

The integral of 36e^{6x}sqrt(5+e^{6x)} is 4(e^{6x}+5)^{3/2}+C

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integral of 36e^{6x}sqrt(5+e^{6x)}

\int 36 e^{6 x} 5 + e^{6 x} d x

4 (e^{6 x} + 5)^{\frac{3}{2}} + C

Solution steps

\int 36 e^{6 x} 5 + e^{6 x} d x

Take the constant out:

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

= 36 \cdot \int e^{6 x} 5 + e^{6 x} d x

Apply u-substitution

: \int \frac{u}{6} d u

= 36 \cdot \int \frac{u}{6} d u

Take the constant out:

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

= 36 \cdot \frac{1}{6} \cdot \int u d u

Apply the Power Rule

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

= 36 \cdot \frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 5 + e^{6 x}

= 36 \cdot \frac{1}{6} \cdot \frac{2}{3} (5 + e^{6 x})^{\frac{3}{2}}

Simplify

36 \cdot \frac{1}{6} \cdot \frac{2}{3} (5 + e^{6 x})^{\frac{3}{2}} : 4 (e^{6 x} + 5)^{\frac{3}{2}}

= 4 (e^{6 x} + 5)^{\frac{3}{2}}

Add a constant to the solution

= 4 (e^{6 x} + 5)^{\frac{3}{2}} + C

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What is the integral of 36e^{6x}sqrt(5+e^{6x)} ?
The integral of 36e^{6x}sqrt(5+e^{6x)} is 4(e^{6x}+5)^{3/2}+C