Solve ∫ (from 0 to 3) of (10x^4)x

$$\int _ { 0 } ^ { 3 } ( 10 x ^ { 4 } ) x$$

Evaluate

$$1215$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\int _{0}^{3}10x^{5}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 10x^{5}\mathrm{d}x$$

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

$$10\int x^{5}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 10 times \frac{x^{6}}{6}.

$$\frac{5x^{6}}{3}$$

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.

$$\frac{5}{3}\times 3^{6}-\frac{5}{3}\times 0^{6}$$

Simplify.

$$1215$$