Evaluate the Integral

Problem

(∫_1^2)((t+1/t)2*d(t))

Solution

1. Expand the integrand using the square of a binomial formula (a+b)2=a2+2*a*b+b2

(t+1/t)2=t2+2*(t)*(1/t)+(1/t)2

=t2+2+t(−2)

2. Rewrite the integral with the expanded terms.

(∫_1^2)((t2+2+t(−2))*d(t))

3. Apply the power rule for integration (∫_^)(tn*d(t))=(t(n+1))/(n+1) to each term.

(∫_^)(t2*d(t))=(t3)/3

(∫_^)(2*d(t))=2*t

(∫_^)(t(−2)*d(t))=(t(−1))/(−1)=−1/t

4. Evaluate the definite integral by substituting the upper limit 2 and the lower limit 1

[(t3)/3+2*t−1/t]21

5. Substitute the upper limit t=2

(2/3+2*(2)−1/2)=8/3+4−1/2

16/6+24/6−3/6=37/6

6. Substitute the lower limit t=1

(1/3+2*(1)−1/1)=1/3+2−1

1/3+1=4/3

7. Subtract the lower limit value from the upper limit value.

37/6−4/3=37/6−8/6

29/6

Final Answer

(∫_1^2)((t+1/t)2*d(t))=29/6

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