Solve ∫ (2x^2-cosx)

$$\int ( 2 x ^ { 2 } - \cos x )$$

Evaluate

$$-\sin(x)+\frac{2x^{3}}{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int -\cos(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x-\int \cos(x)\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}-\int \cos(x)\mathrm{d}x$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$\frac{2x^{3}}{3}-\sin(x)$$

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.

$$\frac{2x^{3}}{3}-\sin(x)+С$$

Differentiate w.r.t. x

$$-\cos(x)+2x^{2}$$