Solve ∫ 2^-3+x^4 wrt x

$$\int 2 ^ { - 3 } + x ^ { 4 } d x$$

Evaluate

$$\frac{x^{5}}{5}+\frac{x}{8}+С$$

Solution Steps

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\int \frac{1}{8}+x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{1}{8}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

Find the integral of \frac{1}{8} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x}{8}+\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.

$$\frac{x}{8}+\frac{x^{5}}{5}$$

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.

$$\frac{x}{8}+\frac{x^{5}}{5}+С$$

Differentiate w.r.t. x

$$x^{4}+\frac{1}{8}$$