What is the integral of (1+30x)sqrt(x) ?

The integral of (1+30x)sqrt(x) is 2/3 x^{3/2}+12x^{5/2}+C

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integral of (1+30x)sqrt(x)

\int (1 + 30 x) x d x

\frac{2}{3} x^{\frac{3}{2}} + 12 x^{\frac{5}{2}} + C

Solution steps

\int (1 + 30 x) x d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} (1 + 30 x) - \int 20 x^{\frac{3}{2}} d x

= \frac{2}{3} x^{\frac{3}{2}} (1 + 30 x) - \int 20 x^{\frac{3}{2}} d x

\int 20 x^{\frac{3}{2}} d x = 8 x^{\frac{5}{2}}

= \frac{2}{3} x^{\frac{3}{2}} (1 + 30 x) - 8 x^{\frac{5}{2}}

Simplify

\frac{2}{3} x^{\frac{3}{2}} (1 + 30 x) - 8 x^{\frac{5}{2}} : \frac{2}{3} x^{\frac{3}{2}} + 12 x^{\frac{5}{2}}

= \frac{2}{3} x^{\frac{3}{2}} + 12 x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 12 x^{\frac{5}{2}} + C

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What is the integral of (1+30x)sqrt(x) ?
The integral of (1+30x)sqrt(x) is 2/3 x^{3/2}+12x^{5/2}+C