What is the integral of 4xy(x^2+y^2)^{0.5} ?

The integral of 4xy(x^2+y^2)^{0.5} is (4y(y^2+x^2)^{3/2})/3+C

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integral of 4xy(x^2+y^2)^{0.5}

\int 4 x y (x^{2} + y^{2})^{0.5} d x

\frac{4 y ( y ^{2} + x ^{2} ) ^{\frac{3}{2}}}{3} + C

Solution steps

\int 4 x y (x^{2} + y^{2})^{0.5} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 y \cdot \int (y^{2} + x^{2})^{0.5} x d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{2}}}{2} d u

= 4 y \cdot \int \frac{u ^{\frac{1}{2}}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 y \frac{1}{2} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 4 y \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = y^{2} + x^{2}

= 4 y \frac{1}{2} \cdot \frac{2}{3} (y^{2} + x^{2})^{\frac{3}{2}}

Simplify

4 y \frac{1}{2} \cdot \frac{2}{3} (y^{2} + x^{2})^{\frac{3}{2}} : \frac{4 y ( y ^{2} + x ^{2} ) ^{\frac{3}{2}}}{3}

= \frac{4 y ( y ^{2} + x ^{2} ) ^{\frac{3}{2}}}{3}

Add a constant to the solution

= \frac{4 y ( y ^{2} + x ^{2} ) ^{\frac{3}{2}}}{3} + C

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What is the integral of 4xy(x^2+y^2)^{0.5} ?
The integral of 4xy(x^2+y^2)^{0.5} is (4y(y^2+x^2)^{3/2})/3+C