Solve ∫ 2x+10sinx wrt x

$$\int 2 x + 10 \sin x d x$$

Evaluate

$$-10\cos(x)+x^{2}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int 10\sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\left(\int x\mathrm{d}x+5\int \sin(x)\mathrm{d}x\right)$$

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}.

$$x^{2}+10\int \sin(x)\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 10 times -\cos(x).

$$x^{2}-10\cos(x)$$

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.

$$x^{2}-10\cos(x)+С$$

Differentiate w.r.t. x

$$2\left(5\sin(x)+x\right)$$