Solve ∫ 2x^2+x^1-15/x+3 wrt x

$$\int{ \frac{ 2 { x }^{ 2 } + { x }^{ 1 } -15 }{ x+3 } }d x$$

Evaluate

$$x^{2}-5x+С$$

Solution Steps

Calculate x to the power of 1 and get x.

$$\int \frac{2x^{2}+x-15}{x+3}\mathrm{d}x$$

Factor the expressions that are not already factored in \frac{2x^{2}+x-15}{x+3}.

$$\int \frac{\left(2x-5\right)\left(x+3\right)}{x+3}\mathrm{d}x$$

Cancel out x+3 in both numerator and denominator.

$$\int 2x-5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x+\int -5\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 2 times \frac{x^{2}}{2}.

$$x^{2}+\int -5\mathrm{d}x$$

Find the integral of -5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{2}-5x$$

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.

$$-5x+x^{2}+С$$

Differentiate w.r.t. x

$$2x-5$$