Solve ∫ 10^x+3sin(x)-sqrt{x wrt x

$$\int{ { 10 }^{ x } +3 \sin ( x ) - \sqrt{ x } }d x$$

Evaluate

$$\log(e)\times 10^{x}-3\cos(x)-\frac{2x^{\frac{3}{2}}}{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 10^{x}\mathrm{d}x+\int 3\sin(x)\mathrm{d}x+\int -\sqrt{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 10^{x}\mathrm{d}x+3\int \sin(x)\mathrm{d}x-\int \sqrt{x}\mathrm{d}x$$

Use \int a^{b}\mathrm{d}b=\frac{a^{b}}{\ln(a)} from the table of common integrals to obtain the result.

$$\frac{10^{x}}{\ln(10)}+3\int \sin(x)\mathrm{d}x-\int \sqrt{x}\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 3 times -\cos(x).

$$\frac{10^{x}}{\ln(10)}-3\cos(x)-\int \sqrt{x}\mathrm{d}x$$

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. Multiply -1 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{10^{x}}{\ln(10)}-3\cos(x)-\frac{2x^{\frac{3}{2}}}{3}$$

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.

$$\frac{10^{x}}{\ln(10)}-3\cos(x)-\frac{2x^{\frac{3}{2}}}{3}+С$$

Differentiate w.r.t. x

$$10^{x}+3\sin(x)-\sqrt{x}$$