Solve ∫ x(3x^2+4)5 wrt x

$$\int x ( 3 x ^ { 2 } + 4 ) 5 d x$$

Evaluate

$$\frac{15x^{4}}{4}+10x^{2}+С$$

Solution Steps

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

$$\int \left(3x^{3}+4x\right)\times 5\mathrm{d}x$$

Use the distributive property to multiply 3x^{3}+4x by 5.

$$\int 15x^{3}+20x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 15x^{3}\mathrm{d}x+\int 20x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$15\int x^{3}\mathrm{d}x+20\int x\mathrm{d}x$$

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}. Multiply 15 times \frac{x^{4}}{4}.

$$\frac{15x^{4}}{4}+20\int x\mathrm{d}x$$

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

$$\frac{15x^{4}}{4}+10x^{2}$$

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.

$$\frac{15x^{4}}{4}+10x^{2}+С$$

Differentiate w.r.t. x

$$15x^{3}+20x$$