What is the integral of x^{1/3}(2x+3) ?

The integral of x^{1/3}(2x+3) is 6/7 x^{7/3}+9/4 x^{4/3}+C

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integral of x^{1/3}(2x+3)

\int x^{\frac{1}{3}} (2 x + 3) d x

\frac{6}{7} x^{\frac{7}{3}} + \frac{9}{4} x^{\frac{4}{3}} + C

Solution steps

\int x^{\frac{1}{3}} (2 x + 3) d x

Expand

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

= \int 2 x^{\frac{4}{3}} + 3 x^{\frac{1}{3}} d x

Apply the Sum Rule:

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

= \int 2 x^{\frac{4}{3}} d x + \int 3 x^{\frac{1}{3}} d x

\int 2 x^{\frac{4}{3}} d x = \frac{6}{7} x^{\frac{7}{3}}

\int 3 x^{\frac{1}{3}} d x = \frac{9}{4} x^{\frac{4}{3}}

= \frac{6}{7} x^{\frac{7}{3}} + \frac{9}{4} x^{\frac{4}{3}}

Add a constant to the solution

= \frac{6}{7} x^{\frac{7}{3}} + \frac{9}{4} x^{\frac{4}{3}} + C

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What is the integral of x^{1/3}(2x+3) ?
The integral of x^{1/3}(2x+3) is 6/7 x^{7/3}+9/4 x^{4/3}+C