What is the integral of (e^{4x}-4)^{1/5}*8e^{4x} ?

The integral of (e^{4x}-4)^{1/5}*8e^{4x} is 5/3 (e^{4x}-4)^{6/5}+C

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integral of (e^{4x}-4)^{1/5}*8e^{4x}

\int (e^{4 x} - 4)^{\frac{1}{5}} \cdot 8 e^{4 x} d x

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

Solution steps

\int (e^{4 x} - 4)^{\frac{1}{5}} \cdot 8 e^{4 x} d x

Take the constant out:

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

= 8 \cdot \int (e^{4 x} - 4)^{\frac{1}{5}} e^{4 x} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{5}}}{4} d u

= 8 \cdot \int \frac{u ^{\frac{1}{5}}}{4} d u

Take the constant out:

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

= 8 \cdot \frac{1}{4} \cdot \int u^{\frac{1}{5}} d u

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= 8 \cdot \frac{1}{4} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = e^{4 x} - 4

= 8 \cdot \frac{1}{4} \cdot \frac{5}{6} (e^{4 x} - 4)^{\frac{6}{5}}

Simplify

8 \cdot \frac{1}{4} \cdot \frac{5}{6} (e^{4 x} - 4)^{\frac{6}{5}} : \frac{5}{3} (e^{4 x} - 4)^{\frac{6}{5}}

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

Add a constant to the solution

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

\frac{2 x}{1 - cot ( x )}

x \to 0 lim (\frac{1}{x + \frac{1}{x}})

\frac{8 s}{3 ( s ^{2} - 16 ) ^{2}}

3 y + x^{4} y^{^{'}} = 2 x y

(2, 2), (- 2, - 2)

What is the integral of (e^{4x}-4)^{1/5}*8e^{4x} ?
The integral of (e^{4x}-4)^{1/5}*8e^{4x} is 5/3 (e^{4x}-4)^{6/5}+C