Solve ∫ x^-1(3x^2+4x) wrt x

$$\int{ { x }^{ -1 } (3 { x }^{ 2 } +4x) }d x$$

Evaluate

$$\frac{3x^{2}}{2}+4x+С$$

Solution Steps

Use the distributive property to multiply x^{-1} by 3x^{2}+4x.

$$\int 3x^{-1}x^{2}+4x^{-1}x\mathrm{d}x$$

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

$$\int 3x^{1}+4x^{-1}x\mathrm{d}x$$

Multiply x^{-1} and x to get 1.

$$\int 3x^{1}+4\mathrm{d}x$$

Calculate x to the power of 1 and get x.

$$\int 3x+4\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x\mathrm{d}x+\int 4\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x\mathrm{d}x+\int 4\mathrm{d}x$$

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 3 times \frac{x^{2}}{2}.

$$\frac{3x^{2}}{2}+\int 4\mathrm{d}x$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3x^{2}}{2}+4x$$

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.

$$\frac{3x^{2}}{2}+4x+С$$

Differentiate w.r.t. x

$$3x+4$$