What is the integral of (5e^{4x})/(sqrt(e^{4x)+3)} ?

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

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integral of (5e^{4x})/(sqrt(e^{4x)+3)}

\int \frac{5 e ^{4 x}}{e ^{4 x} + 3} d x

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

Solution steps

\int \frac{5 e ^{4 x}}{e ^{4 x} + 3} d x

Take the constant out:

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

= 5 \cdot \int \frac{e ^{4 x}}{e ^{4 x} + 3} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= 5 \cdot \int \frac{1}{4 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 5 \cdot \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = e^{4 x} + 3

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

Simplify

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

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

Add a constant to the solution

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

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