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

The integral of (x^{2/3}+1)^2 is 3/7 x^{7/3}+6/5 x^{5/3}+x+C

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

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

\frac{3}{7} x^{\frac{7}{3}} + \frac{6}{5} x^{\frac{5}{3}} + x + C

Solution steps

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

Expand

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

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

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

\int 2 x^{\frac{2}{3}} d x = \frac{6}{5} x^{\frac{5}{3}}

\int 1 d x = x

= \frac{3}{7} x^{\frac{7}{3}} + \frac{6}{5} x^{\frac{5}{3}} + x

Add a constant to the solution

= \frac{3}{7} x^{\frac{7}{3}} + \frac{6}{5} x^{\frac{5}{3}} + x + C

What is the integral of (x^{2/3}+1)^2 ?
The integral of (x^{2/3}+1)^2 is 3/7 x^{7/3}+6/5 x^{5/3}+x+C