Solve ∫ (sinx+sqrt{x}-e) wrt x

$$\int ( \sin x + \sqrt { x } - e ) d x$$

Evaluate

$$-\cos(x)+\frac{2x^{\frac{3}{2}}}{3}-ex+С$$

Solution Steps

Integrate the sum term by term.

$$\int \sin(x)\mathrm{d}x+\int \sqrt{x}\mathrm{d}x+\int -e\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sin(x)\mathrm{d}x+\int \sqrt{x}\mathrm{d}x-\int e\mathrm{d}x$$

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

$$-\cos(x)+\int \sqrt{x}\mathrm{d}x-\int e\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify.

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

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

$$-\cos(x)+\frac{2x^{\frac{3}{2}}}{3}-ex$$

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.

$$-\cos(x)+\frac{2x^{\frac{3}{2}}}{3}-ex+С$$

Differentiate w.r.t. x

$$\sin(x)+\sqrt{x}-e$$