What is the integral of e^{2y}(4-e^{2y})^{4/3} ?

The integral of e^{2y}(4-e^{2y})^{4/3} is -3/14 (4-e^{2y})^{7/3}+C

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integral of e^{2y}(4-e^{2y})^{4/3}

\int e^{2 y} (4 - e^{2 y})^{\frac{4}{3}} d y

- \frac{3}{14} (4 - e^{2 y})^{\frac{7}{3}} + C

Solution steps

\int e^{2 y} (4 - e^{2 y})^{\frac{4}{3}} d y

Apply u-substitution

: \int - \frac{u ^{\frac{4}{3}}}{2} d u

= \int - \frac{u ^{\frac{4}{3}}}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u^{\frac{4}{3}} d u

Apply the Power Rule

: \frac{3}{7} u^{\frac{7}{3}}

= - \frac{1}{2} \cdot \frac{3}{7} u^{\frac{7}{3}}

Substitute back

u = 4 - e^{2 y}

= - \frac{1}{2} \cdot \frac{3}{7} (4 - e^{2 y})^{\frac{7}{3}}

Simplify

- \frac{1}{2} \cdot \frac{3}{7} (4 - e^{2 y})^{\frac{7}{3}} : - \frac{3}{14} (4 - e^{2 y})^{\frac{7}{3}}

= - \frac{3}{14} (4 - e^{2 y})^{\frac{7}{3}}

Add a constant to the solution

= - \frac{3}{14} (4 - e^{2 y})^{\frac{7}{3}} + C

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What is the integral of e^{2y}(4-e^{2y})^{4/3} ?
The integral of e^{2y}(4-e^{2y})^{4/3} is -3/14 (4-e^{2y})^{7/3}+C