Solve ∫ (from 1 to 2) of 5x^2

$$\int _ { 1 } ^ { 2 } 5 x ^ { 2 }$$

Evaluate

$$\frac{35}{3}\approx 11.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5x^{2}\mathrm{d}x$$

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

$$5\int x^{2}\mathrm{d}x$$

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

$$\frac{5x^{3}}{3}$$

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.

$$\frac{5}{3}\times 2^{3}-\frac{5}{3}\times 1^{3}$$

Simplify.

$$\frac{35}{3}$$