Solve ∫ (from 0 to 2) of (1/3x^4+3/5x^4)

$$\int _ { 0 } ^ { 2 } ( \frac { 1 } { 3 } x ^ { 4 } + \frac { 3 } { 5 } x ^ { 4 } )$$

Evaluate

$$\frac{448}{75}\approx 5.973333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x^{4}}{3}+\frac{3x^{4}}{5}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{x^{4}}{3}\mathrm{d}x+\int \frac{3x^{4}}{5}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{\int x^{4}\mathrm{d}x}{3}+\frac{3\int x^{4}\mathrm{d}x}{5}$$

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}. Multiply \frac{1}{3} times \frac{x^{5}}{5}.

$$\frac{x^{5}}{15}+\frac{3\int x^{4}\mathrm{d}x}{5}$$

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}. Multiply \frac{3}{5} times \frac{x^{5}}{5}.

$$\frac{x^{5}}{15}+\frac{3x^{5}}{25}$$

Simplify.

$$\frac{14x^{5}}{75}$$

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.

$$\frac{14}{75}\times 2^{5}-\frac{14}{75}\times 0^{5}$$

Simplify.

$$\frac{448}{75}$$