Solve ∫ (from 0 to 1) of (5u^2+pi^2)du

$$\int _ { 0 } ^ { 1 } ( 5 u ^ { 2 } + \pi ^ { 2 } ) d u$$

Evaluate

$$\pi ^{2}+\frac{5}{3}\approx 11.536271068$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5u^{2}+\pi ^{2}\mathrm{d}u$$

Integrate the sum term by term.

$$\int 5u^{2}\mathrm{d}u+\int \pi ^{2}\mathrm{d}u$$

Factor out the constant in each of the terms.

$$5\int u^{2}\mathrm{d}u+\int \pi ^{2}\mathrm{d}u$$

Since \int u^{k}\mathrm{d}u=\frac{u^{k+1}}{k+1} for k\neq -1, replace \int u^{2}\mathrm{d}u with \frac{u^{3}}{3}. Multiply 5 times \frac{u^{3}}{3}.

$$\frac{5u^{3}}{3}+\int \pi ^{2}\mathrm{d}u$$

Find the integral of \pi ^{2} using the table of common integrals rule \int a\mathrm{d}u=au.

$$\frac{5u^{3}}{3}+\pi ^{2}u$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{5}{3}\times 1^{3}+\pi ^{2}\times 1-\left(\frac{5}{3}\times 0^{3}+\pi ^{2}\times 0\right)$$

Simplify.

$$\frac{5}{3}+\pi ^{2}$$