Solve ∫ (from 1 to 2) of 2(t^9-2t^7+t^5)dt

$$\int _ { 1 } ^ { 2 } 2 ( t ^ { 9 } - 2 t ^ { 7 } + t ^ { 5 } ) d t$$

Evaluate

$$\frac{981}{10}=98.1$$

Solution Steps

Use the distributive property to multiply 2 by t^{9}-2t^{7}+t^{5}.

$$\int _{1}^{2}2t^{9}-4t^{7}+2t^{5}\mathrm{d}t$$

Evaluate the indefinite integral first.

$$\int 2t^{9}-4t^{7}+2t^{5}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 2t^{9}\mathrm{d}t+\int -4t^{7}\mathrm{d}t+\int 2t^{5}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$2\int t^{9}\mathrm{d}t-4\int t^{7}\mathrm{d}t+2\int t^{5}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{9}\mathrm{d}t with \frac{t^{10}}{10}. Multiply 2 times \frac{t^{10}}{10}.

$$\frac{t^{10}}{5}-4\int t^{7}\mathrm{d}t+2\int t^{5}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{7}\mathrm{d}t with \frac{t^{8}}{8}. Multiply -4 times \frac{t^{8}}{8}.

$$\frac{t^{10}}{5}-\frac{t^{8}}{2}+2\int t^{5}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{5}\mathrm{d}t with \frac{t^{6}}{6}. Multiply 2 times \frac{t^{6}}{6}.

$$\frac{t^{10}}{5}-\frac{t^{8}}{2}+\frac{t^{6}}{3}$$

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.

$$\frac{2^{10}}{5}-\frac{2^{8}}{2}+\frac{2^{6}}{3}-\left(\frac{1^{10}}{5}-\frac{1^{8}}{2}+\frac{1^{6}}{3}\right)$$

Simplify.

$$\frac{981}{10}$$