Solve ∫ 3cos(θ)+3sqrt[]{θ}}dθ

$$\int{ 3 \cos ( \theta ) +3 \sqrt[ ]{ \theta } }d \theta$$

Evaluate

$$3\sin(\theta )+2\theta ^{\frac{3}{2}}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3\cos(\theta )\mathrm{d}\theta +\int 3\sqrt{\theta }\mathrm{d}\theta$$

Factor out the constant in each of the terms.

$$3\left(\int \cos(\theta )\mathrm{d}\theta +\int \sqrt{\theta }\mathrm{d}\theta \right)$$

Use \int \cos(\theta )\mathrm{d}\theta =\sin(\theta ) from the table of common integrals to obtain the result.

$$3\left(\sin(\theta )+\int \sqrt{\theta }\mathrm{d}\theta \right)$$

Rewrite \sqrt{\theta } as \theta ^{\frac{1}{2}}. Since \int \theta ^{k}\mathrm{d}\theta =\frac{\theta ^{k+1}}{k+1} for k\neq -1, replace \int \theta ^{\frac{1}{2}}\mathrm{d}\theta with \frac{\theta ^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply 3 times \frac{2\theta ^{\frac{3}{2}}}{3}.

$$3\sin(\theta )+2\theta ^{\frac{3}{2}}$$

If F\left(\theta \right) is an antiderivative of f\left(\theta \right), then the set of all antiderivatives of f\left(\theta \right) is given by F\left(\theta \right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$3\sin(\theta )+2\theta ^{\frac{3}{2}}+С$$

Differentiate w.r.t. θ

$$3\left(\cos(\theta )+\sqrt{\theta }\right)$$