Solve ∫ (4xy^2)^2

$$\int ( 4 x y ^ { 2 } ) ^ { 2 }$$

Evaluate

$$\frac{16x^{3}y^{4}}{3}+С$$

Solution Steps

Expand \left(4xy^{2}\right)^{2}.

$$\int 4^{2}x^{2}\left(y^{2}\right)^{2}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int 4^{2}x^{2}y^{4}\mathrm{d}x$$

Calculate 4 to the power of 2 and get 16.

$$\int 16x^{2}y^{4}\mathrm{d}x$$

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

$$16y^{4}\int x^{2}\mathrm{d}x$$

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

$$16y^{4}\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{16y^{4}x^{3}}{3}$$

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

$$\frac{16y^{4}x^{3}}{3}+С$$

Differentiate w.r.t. x

$$16x^{2}y^{4}$$