Solve ∫ int{x^2left(x^2+y^2right)^3 wrt x}dA8

$$\int{ \int{ { x }^{ 2 } { \left( { x }^{ 2 } + { y }^{ 2 } \right) }^{ 3 } }d x }d A 8$$

Evaluate

$$A_{8}\left(\frac{x^{3}y^{6}}{3}+\frac{3y^{2}x^{7}}{7}+\frac{3y^{4}x^{5}}{5}+\frac{x^{9}}{9}\right)+СA_{8}+С_{1}$$

Solution Steps

Find the integral of \int x^{2}\left(x^{2}+y^{2}\right)^{3}\mathrm{d}x using the table of common integrals rule \int a\mathrm{d}A_{8}=aA_{8}.

$$\int x^{2}\left(x^{2}+y^{2}\right)^{3}\mathrm{d}xA_{8}$$

Simplify.

$$\left(\frac{y^{6}x^{3}}{3}+\frac{3y^{4}x^{5}}{5}+\frac{3y^{2}x^{7}}{7}+\frac{x^{9}}{9}+С\right)A_{8}$$

If F\left(A_{8}\right) is an antiderivative of f\left(A_{8}\right), then the set of all antiderivatives of f\left(A_{8}\right) is given by F\left(A_{8}\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\left(\frac{y^{6}x^{3}}{3}+\frac{3y^{4}x^{5}}{5}+\frac{3y^{2}x^{7}}{7}+\frac{x^{9}}{9}+С\right)A_{8}+С$$

Differentiate w.r.t. x

$$A_{8}x^{2}\left(x^{2}+y^{2}\right)^{3}$$