Solve ∫ (from 0 to 3) of 30.7+0.81t^2+0.1215t^4+0.01215t^6dt

$$\int _ { 0 } ^ { 3 } 30.7 + 0.81 t ^ { 2 } + 0.1215 t ^ { 4 } + 0.01215 t ^ { 6 } d t$$

Evaluate

$$\frac{15272727}{140000}\approx 109.090907143$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 30.7+\frac{81t^{2}}{100}+\frac{243t^{4}}{2000}+\frac{243t^{6}}{20000}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 30.7\mathrm{d}t+\int \frac{81t^{2}}{100}\mathrm{d}t+\int \frac{243t^{4}}{2000}\mathrm{d}t+\int \frac{243t^{6}}{20000}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$\int 30.7\mathrm{d}t+\frac{81\int t^{2}\mathrm{d}t}{100}+\frac{243\int t^{4}\mathrm{d}t}{2000}+\frac{243\int t^{6}\mathrm{d}t}{20000}$$

Find the integral of 30.7 using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{307t}{10}+\frac{81\int t^{2}\mathrm{d}t}{100}+\frac{243\int t^{4}\mathrm{d}t}{2000}+\frac{243\int t^{6}\mathrm{d}t}{20000}$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply 0.81 times \frac{t^{3}}{3}.

$$\frac{307t}{10}+\frac{27t^{3}}{100}+\frac{243\int t^{4}\mathrm{d}t}{2000}+\frac{243\int t^{6}\mathrm{d}t}{20000}$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{4}\mathrm{d}t with \frac{t^{5}}{5}. Multiply 0.1215 times \frac{t^{5}}{5}.

$$\frac{307t}{10}+\frac{27t^{3}}{100}+\frac{243t^{5}}{10000}+\frac{243\int t^{6}\mathrm{d}t}{20000}$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{6}\mathrm{d}t with \frac{t^{7}}{7}. Multiply 0.01215 times \frac{t^{7}}{7}.

$$\frac{307t}{10}+\frac{27t^{3}}{100}+\frac{243t^{5}}{10000}+\frac{243t^{7}}{140000}$$

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.

$$30.7\times 3+\frac{27}{100}\times 3^{3}+\frac{243}{10000}\times 3^{5}+\frac{243}{140000}\times 3^{7}-\left(30.7\times 0+\frac{27}{100}\times 0^{3}+\frac{243}{10000}\times 0^{5}+\frac{243}{140000}\times 0^{7}\right)$$

Simplify.

$$\frac{15272727}{140000}$$