Solve ∫ 4x^2left(5x^2-x+3right) wrt x

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

Evaluate

$$4x^{5}-x^{4}+4x^{3}+С$$

Solution Steps

Use the distributive property to multiply 4x^{2} by 5x^{2}-x+3.

$$\int 20x^{4}-4x^{3}+12x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 20x^{4}\mathrm{d}x+\int -4x^{3}\mathrm{d}x+\int 12x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$20\int x^{4}\mathrm{d}x-4\int x^{3}\mathrm{d}x+12\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply 20 times \frac{x^{5}}{5}.

$$4x^{5}-4\int x^{3}\mathrm{d}x+12\int x^{2}\mathrm{d}x$$

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 -4 times \frac{x^{4}}{4}.

$$4x^{5}-x^{4}+12\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 12 times \frac{x^{3}}{3}.

$$4x^{5}-x^{4}+4x^{3}$$

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.

$$4x^{5}-x^{4}+4x^{3}+С$$

Differentiate w.r.t. x

$$4x^{2}\left(5x^{2}-x+3\right)$$