Solve ∫ (θ^2+cosθ)dθ

$$\int ( \theta ^ { 2 } + \cos \theta ) d \theta$$

Evaluate

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

Solution Steps

Integrate the sum term by term.

$$\int \theta ^{2}\mathrm{d}\theta +\int \cos(\theta )\mathrm{d}\theta$$

Since \int \theta ^{k}\mathrm{d}\theta =\frac{\theta ^{k+1}}{k+1} for k\neq -1, replace \int \theta ^{2}\mathrm{d}\theta with \frac{\theta ^{3}}{3}.

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

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

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

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.

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

Differentiate w.r.t. θ

$$\cos(\theta )+\theta ^{2}$$