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

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

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

\int (1 + 30 w) w d w

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

Solution steps

\int (1 + 30 w) w d w

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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