What is the integral of 36e^{4x}sqrt(3+e^{4x)} ?

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

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integral of 36e^{4x}sqrt(3+e^{4x)}

\int 36 e^{4 x} 3 + e^{4 x} d x

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

Solution steps

\int 36 e^{4 x} 3 + e^{4 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^{4 x} 3 + e^{4 x} d x

Apply u-substitution

: \int \frac{u}{4} d u

= 36 \cdot \int \frac{u}{4} d u

Take the constant out:

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

= 36 \cdot \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

= 36 \cdot \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 + e^{4 x}

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

Simplify

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

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

Add a constant to the solution

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

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