Evaluate the Integral

Problem

(∫_^)(x2*cos(2*x3)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 2*x3 is a multiple of the outer factor x2

2. Substitute u=2*x3 to simplify the integrand.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=6*x2

d(u)=6*x2*d(x)

1/6*d(u)=x2*d(x)

4. Rewrite the integral in terms of u by substituting 2*x3 with u and x2*d(x) with 1/6*d(u)

(∫_^)(cos(u)⋅1/6*d(u))

1/6*(∫_^)(cos(u)*d(u))

5. Integrate the function with respect to u

1/6*sin(u)+C

6. Back-substitute the original expression 2*x3 for u to get the final result in terms of x

Final Answer

(∫_^)(x2*cos(2*x3)*d(x))=sin(2*x3)/6+C

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