What is the integral of 2sqrt(x)(1+x^2) ?

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

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integral of 2sqrt(x)(1+x^2)

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

2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}}) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int x (1 + x^{2}) d x

Expand

x (1 + x^{2}) : x + x^{\frac{5}{2}}

= 2 \cdot \int x + x^{\frac{5}{2}} 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

= 2 (\int x d x + \int x^{\frac{5}{2}} d x)

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= 2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}})

Add a constant to the solution

= 2 (\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{7} x^{\frac{7}{2}}) + C

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