Evaluate the Integral

Problem

(∫_1^2)((2*u2+3)*d(u))

Solution

1. Identify the integral as a definite integral of a polynomial function with respect to u

2. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1) to each term in the integrand.

3. Find the antiderivative of the expression 2*u2+3

(∫_^)((2*u2+3)*d(u))=(2*u3)/3+3*u

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and the lower limit 1

[(2*u3)/3+3*u]21

5. Substitute the upper limit u=2 into the expression.

(2*(2)3)/3+3*(2)=16/3+6

6. Substitute the lower limit u=1 into the expression.

(2*(1)3)/3+3*(1)=2/3+3

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

(16/3+6)−(2/3+3)

8. Simplify the resulting numerical expression.

14/3+3

9. Find a common denominator to combine the terms.

14/3+9/3=23/3

Final Answer

(∫_1^2)((2*u2+3)*d(u))=23/3

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