Solve ∫ -6x^4y^2:(-xy)+2x^3y-(-xy)(-5x^2)

$$\int - 6 x ^ { 4 } y ^ { 2 } : ( - x y ) + 2 x ^ { 3 } y - ( - x y ) ( - 5 x ^ { 2 } )$$

Evaluate

$$\frac{3yx^{4}}{4}+С$$

Solution Steps

Cancel out y in both numerator and denominator.

$$\int \frac{-6yx^{4}}{-x}+2x^{3}y-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{3}y times \frac{-x}{-x}.

$$\int \frac{-6yx^{4}}{-x}+\frac{2x^{3}y\left(-1\right)x}{-x}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

Since \frac{-6yx^{4}}{-x} and \frac{2x^{3}y\left(-1\right)x}{-x} have the same denominator, add them by adding their numerators.

$$\int \frac{-6yx^{4}+2x^{3}y\left(-1\right)x}{-x}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

Do the multiplications in -6yx^{4}+2x^{3}y\left(-1\right)x.

$$\int \frac{-6yx^{4}-2x^{4}y}{-x}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

Combine like terms in -6yx^{4}-2x^{4}y.

$$\int \frac{-8yx^{4}}{-x}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

Cancel out x in both numerator and denominator.

$$\int \frac{-8yx^{3}}{-1}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

Anything divided by -1 gives its opposite.

$$\int 8yx^{3}-\left(-x\right)y\left(-5\right)x^{2}\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\int 8yx^{3}-\left(-x^{3}y\left(-5\right)\right)\mathrm{d}x$$

Multiply -1 and -1 to get 1.

$$\int 8yx^{3}+x^{3}y\left(-5\right)\mathrm{d}x$$

Combine 8yx^{3} and x^{3}y\left(-5\right) to get 3yx^{3}.

$$\int 3yx^{3}\mathrm{d}x$$

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

$$3y\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

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

Simplify.

$$\frac{3yx^{4}}{4}$$

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.

$$\frac{3yx^{4}}{4}+С$$

Differentiate w.r.t. x

$$3yx^{3}$$