Solve ∫ (sqrt{x}+8^3sqrt{x}) wrt x

$$\int ( \sqrt { x } + 8 ^ { 3 } \sqrt { x } ) d x$$

Evaluate

$$342x^{\frac{3}{2}}+С$$

Solution Steps

Calculate 8 to the power of 3 and get 512.

$$\int \sqrt{x}+512\sqrt{x}\mathrm{d}x$$

Combine \sqrt{x} and 512\sqrt{x} to get 513\sqrt{x}.

$$\int 513\sqrt{x}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$513\int \sqrt{x}\mathrm{d}x$$

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 513 times \frac{2x^{\frac{3}{2}}}{3}.

$$342x^{\frac{3}{2}}$$

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.

$$342x^{\frac{3}{2}}+С$$

Differentiate w.r.t. x

$$513\sqrt{x}$$