What is the integral of (3x+2)/(\sqrt[5]{4x-3)} ?

The integral of (3x+2)/(\sqrt[5]{4x-3)} is 1/16 (5/3 (4x-3)^{9/5}+85/4 (4x-3)^{4/5})+C

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integral of (3x+2)/(\sqrt[5]{4x-3)}

\frac{1}{16} (\frac{5}{3} (4 x - 3)^{\frac{9}{5}} + \frac{85}{4} (4 x - 3)^{\frac{4}{5}}) + C

Solution steps

Apply u-substitution

Take the constant out:

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

Expand

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

\int 3 u^{\frac{4}{5}} d u = \frac{5}{3} u^{\frac{9}{5}}

= \frac{1}{16} (\frac{5}{3} u^{\frac{9}{5}} + \frac{85}{4} u^{\frac{4}{5}})

Substitute back

u = 4 x - 3

= \frac{1}{16} (\frac{5}{3} (4 x - 3)^{\frac{9}{5}} + \frac{85}{4} (4 x - 3)^{\frac{4}{5}})

Add a constant to the solution

= \frac{1}{16} (\frac{5}{3} (4 x - 3)^{\frac{9}{5}} + \frac{85}{4} (4 x - 3)^{\frac{4}{5}}) + C

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What is the integral of (3x+2)/(\sqrt[5]{4x-3)} ?
The integral of (3x+2)/(\sqrt[5]{4x-3)} is 1/16 (5/3 (4x-3)^{9/5}+85/4 (4x-3)^{4/5})+C