Evaluate the Integral

Problem

(∫_^)((x6−3*x2)4*(x5−x)*d(x))

Solution

1. Identify a potential substitution by looking for a function and its derivative within the integrand.

2. Define the variable u as the expression inside the power.

u=x6−3*x2

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=6*x5−6*x

d(u)=(6*x5−6*x)*d(x)

4. Factor the expression for d(u) to match the remaining part of the integrand.

d(u)=6*(x5−x)*d(x)

1/6*d(u)=(x5−x)*d(x)

5. Substitute u and 1/6*d(u) into the original integral.

(∫_^)(u4⋅1/6*d(u))

6. Apply the constant multiple rule and the power rule for integration.

1/6*(∫_^)(u4*d(u))

1/6⋅(u5)/5+C

7. Simplify the resulting expression.

(u5)/30+C

8. Back-substitute the original expression for u to get the final answer in terms of x

((x6−3*x2)5)/30+C

Final Answer

(∫_^)((x6−3*x2)4*(x5−x)*d(x))=((x6−3*x2)5)/30+C

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