Solve ∫ (1+2x^4) wrt x

$$\int ( 1 + 2 x ^ { 4 } ) d x$$

Evaluate

$$\frac{2x^{5}}{5}+x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int 2x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x+2\int x^{4}\mathrm{d}x$$

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

$$x+2\int x^{4}\mathrm{d}x$$

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}. Multiply 2 times \frac{x^{5}}{5}.

$$x+\frac{2x^{5}}{5}$$

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.

$$x+\frac{2x^{5}}{5}+С$$

Differentiate w.r.t. x

$$2x^{4}+1$$