Solve ∫ (from 1 to 27) of 2pix(27x^3) wrt x

$$\int _ { 1 } ^ { 27 } 2 \pi x ( 27 x ^ { 3 } ) d x$$

Evaluate

$$\frac{774840924\pi }{5}\approx 486846910.90782547$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\int _{1}^{27}2\pi x^{4}\times 27\mathrm{d}x$$

Multiply 2 and 27 to get 54.

$$\int _{1}^{27}54\pi x^{4}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 54\pi x^{4}\mathrm{d}x$$

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

$$54\pi \int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.

$$54\pi \times \frac{x^{5}}{5}$$

Simplify.

$$\frac{54\pi x^{5}}{5}$$

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.

$$\frac{54}{5}\pi \times 27^{5}-\frac{54}{5}\pi \times 1^{5}$$

Simplify.

$$\frac{774840924\pi }{5}$$