Solve ∫ intx^2(x^2+y^2)^3dA

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

Evaluate

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

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=aA.

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

Simplify.

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

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

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

Differentiate w.r.t. x

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