Solve ∫ x^3-3x^2+7x/8x+19 wrt y

$$\int{ \frac{ { x }^{ 3 } -3 { x }^{ 2 } +7x }{ 8x+19 } }d y$$

Evaluate

$$\frac{7xy+yx^{3}-3yx^{2}}{8x+19}+С$$

Solution Steps

Find the integral of \frac{x^{3}-3x^{2}+7x}{8x+19} using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\frac{x^{3}-3x^{2}+7x}{8x+19}y$$

Simplify.

$$\frac{\left(x^{3}-3x^{2}+7x\right)y}{8x+19}$$

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

$$\frac{\left(x^{3}-3x^{2}+7x\right)y}{8x+19}+С$$

Differentiate w.r.t. x

$$\frac{y\left(16x^{3}+33x^{2}-114x+133\right)}{\left(8x+19\right)^{2}}$$