Evaluate the Integral

Problem

(∫_1^2)(x*sin(x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x2 is a multiple of the outer factor x

2. Substitute u=x2 to simplify the integrand.

3. Differentiate the substitution to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

4. Change the limits of integration from x to u when x=1 u=1=1 when x=2 u=2=4

5. Rewrite the integral in terms of u

(∫_1^4)(1/2*sin(u)*d(u))

6. Integrate the function with respect to u

1/2*[−cos(u)]41

7. Evaluate the definite integral by plugging in the upper and lower limits:

1/2*(−cos(4)−(−cos(1)))

8. Simplify the expression to reach the final result:

1/2*(cos(1)−cos(4))

Final Answer

(∫_1^2)(x*sin(x2)*d(x))=(cos(1)−cos(4))/2

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