Solve ∫ (a^2+2b)da

$$\int ( a ^ { 2 } + 2 b ) d a$$

Evaluate

$$\frac{a^{3}}{3}+2ab+С$$

Solution Steps

Integrate the sum term by term.

$$\int a^{2}\mathrm{d}a+\int 2b\mathrm{d}a$$

Factor out the constant in each of the terms.

$$\int a^{2}\mathrm{d}a+2\int b\mathrm{d}a$$

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

$$\frac{a^{3}}{3}+2\int b\mathrm{d}a$$

Find the integral of b using the table of common integrals rule \int a\mathrm{d}a=aa.

$$\frac{a^{3}}{3}+2ba$$

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

$$\frac{a^{3}}{3}+2ba+С$$

Differentiate w.r.t. a

$$a^{2}+2b$$