Solve ∫ 23^x+4/sqrt{x}+e^7 wrt x

$$\int 23 ^ { x } + \frac { 4 } { \sqrt { x } } + e ^ { 7 } d x$$

Evaluate

$$\frac{23^{x}}{\ln(23)}+e^{7}x+8\sqrt{x}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 23^{x}\mathrm{d}x+\int \frac{4}{\sqrt{x}}\mathrm{d}x+\int e^{7}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 23^{x}\mathrm{d}x+4\int \frac{1}{\sqrt{x}}\mathrm{d}x+\int e^{7}\mathrm{d}x$$

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

$$\frac{23^{x}}{\ln(23)}+4\int \frac{1}{\sqrt{x}}\mathrm{d}x+\int e^{7}\mathrm{d}x$$

Rewrite \frac{1}{\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{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form. Multiply 4 times 2\sqrt{x}.

$$\frac{23^{x}}{\ln(23)}+8\sqrt{x}+\int e^{7}\mathrm{d}x$$

Find the integral of e^{7} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{23^{x}}{\ln(23)}+8\sqrt{x}+e^{7}x$$

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.

$$\frac{23^{x}}{\ln(23)}+8\sqrt{x}+e^{7}x+С$$

Differentiate w.r.t. x

$$23^{x}+e^{7}+\frac{4}{\sqrt{x}}$$