Evaluate the Integral

Problem

(∫_1^2)(3*x5−2*x3*d(x))

Solution

1. Identify the integrand 3*x5−2*x3 and the limits of integration from 1 to 2

2. Apply the power rule for integration, which states (∫_^)(xn*d(x))=(x(n+1))/(n+1) to find the antiderivative.

3. Find the antiderivative of each term:

(∫_^)(3*x5*d(x))=(3*x6)/6=1/2*x6

(∫_^)(2*x3*d(x))=(2*x4)/4=1/2*x4

4. Combine the terms to get the full antiderivative F(x)

F(x)=1/2*x6−1/2*x4

5. Apply the Fundamental Theorem of Calculus by evaluating F(2)−F(1)

F(2)=1/2*(2)6−1/2*(2)4=64/2−16/2=32−8=24

F(1)=1/2*(1)6−1/2*(1)4=1/2−1/2=0

6. Subtract the values to find the final result:

24−0=24

Final Answer

(∫_1^2)(3*x5−2*x3*d(x))=24

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