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

The integral of (5x)/(\sqrt[4]{x+2)} is 5(4/7 (x+2)^{7/4}-8/3 (x+2)^{3/4})+C

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

5 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}}) + C

Solution steps

Take the constant out:

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

Apply u-substitution

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 u^{\frac{3}{4}} d u = \frac{4}{7} u^{\frac{7}{4}}

= 5 (\frac{4}{7} u^{\frac{7}{4}} - \frac{8}{3} u^{\frac{3}{4}})

Substitute back

u = x + 2

= 5 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}})

Add a constant to the solution

= 5 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}}) + C

What is the integral of (5x)/(\sqrt[4]{x+2)} ?
The integral of (5x)/(\sqrt[4]{x+2)} is 5(4/7 (x+2)^{7/4}-8/3 (x+2)^{3/4})+C