What is the integral of sqrt(y)(2y^2-3) ?

The integral of sqrt(y)(2y^2-3) is 4/7 y^{7/2}-2y^{3/2}+C

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integral of sqrt(y)(2y^2-3)

\int y (2 y^{2} - 3) d y

\frac{4}{7} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} + C

Solution steps

\int y (2 y^{2} - 3) d y

Apply Integration By Parts

: \frac{4}{3} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} - \int \frac{8}{3} y^{\frac{5}{2}} d y

= \frac{4}{3} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} - \int \frac{8}{3} y^{\frac{5}{2}} d y

\int \frac{8}{3} y^{\frac{5}{2}} d y = \frac{16}{21} y^{\frac{7}{2}}

= \frac{4}{3} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} - \frac{16}{21} y^{\frac{7}{2}}

Simplify

\frac{4}{3} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} - \frac{16}{21} y^{\frac{7}{2}} : \frac{4}{7} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}}

= \frac{4}{7} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{7} y^{\frac{7}{2}} - 2 y^{\frac{3}{2}} + C

\frac{d}{d x} (4^{- 2 x})

\frac{d ^{2}}{d x ^{2}} y + 81 y = 0

\frac{d}{d x} (\frac{1 - cosh ( x )}{1 + cosh ( x )})

\int_{0}^{1} x^{2} (x^{3} + 1)^{3} d x

\frac{d}{d x} ((e^{x})^{x})

What is the integral of sqrt(y)(2y^2-3) ?
The integral of sqrt(y)(2y^2-3) is 4/7 y^{7/2}-2y^{3/2}+C