Solve ∫ 4pi/3sqrt{y^3 wrt y=

$$\int \frac { 4 \pi } { 3 } \sqrt { y ^ { 3 } } d y =$$

Evaluate

$$\frac{8\pi y^{\frac{5}{2}}}{15}+С$$

Solution Steps

Factor out the constant using \int af\left(y\right)\mathrm{d}y=a\int f\left(y\right)\mathrm{d}y.

$$\frac{4\pi }{3}\int y^{\frac{3}{2}}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{\frac{3}{2}}\mathrm{d}y with \frac{2y^{\frac{5}{2}}}{5}.

$$\frac{4\pi }{3}\times \frac{2y^{\frac{5}{2}}}{5}$$

Simplify.

$$\frac{8\pi y^{\frac{5}{2}}}{15}$$

If F\left(y\right) is an antiderivative of f\left(y\right), then the set of all antiderivatives of f\left(y\right) is given by F\left(y\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{8\pi y^{\frac{5}{2}}}{15}+С$$

Differentiate w.r.t. y

$$\frac{4\pi y^{\frac{3}{2}}}{3}$$