Solve ∫ (from 272 to 800) of (30+10.7*10^-3T)dT

$$\int _ { 272 } ^ { 800 } ( 30 + 10.7 \times 10 ^ { - 3 } T ) d T$$

Evaluate

$$18868.1856$$

Solution Steps

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\int _{272}^{800}30+10.7\times \frac{1}{1000}T\mathrm{d}T$$

Multiply 10.7 and \frac{1}{1000} to get \frac{107}{10000}.

$$\int _{272}^{800}30+\frac{107}{10000}T\mathrm{d}T$$

Evaluate the indefinite integral first.

$$\int 30+\frac{107T}{10000}\mathrm{d}T$$

Integrate the sum term by term.

$$\int 30\mathrm{d}T+\int \frac{107T}{10000}\mathrm{d}T$$

Factor out the constant in each of the terms.

$$\int 30\mathrm{d}T+\frac{107\int T\mathrm{d}T}{10000}$$

Find the integral of 30 using the table of common integrals rule \int a\mathrm{d}T=aT.

$$30T+\frac{107\int T\mathrm{d}T}{10000}$$

Since \int T^{k}\mathrm{d}T=\frac{T^{k+1}}{k+1} for k\neq -1, replace \int T\mathrm{d}T with \frac{T^{2}}{2}. Multiply \frac{107}{10000} times \frac{T^{2}}{2}.

$$30T+\frac{107T^{2}}{20000}$$

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.

$$30\times 800+\frac{107}{20000}\times 800^{2}-\left(30\times 272+\frac{107}{20000}\times 272^{2}\right)$$

Simplify.

$$\frac{11792616}{625}$$