Evaluate the Integral

Problem

(∫_0^√(,π))(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. Define the substitution variable u=x2

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

4. Determine the new limits of integration by substituting the original x values into the equation u=x2 when x=0 u=0=0 when x=√(,π) u=(√(,π))2=π

5. Rewrite the integral in terms of u

(∫_0^π)(1/2*sin(u)*d(u))

6. Integrate the function with respect to u

1/2*[−cos(u)]π0

7. Evaluate the definite integral at the boundaries:

1/2*(−cos(π)−(−cos(0)))

8. Simplify the trigonometric values where cos(π)=−1 and cos(0)=1

1/2*(−(−1)−(−1))

1/2*(1+1)

1

Final Answer

(∫_0^√(,π))(x*sin(x2)*d(x))=1

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