Evaluate the Integral integral of (4-5x)^6 with respect to x

Problem

(∫_^)((4−5*x)6*d(x))

Solution

1. Identify the integral as a power of a linear function, which suggests using usubstitution.

2. Substitute u=4−5*x which implies that the derivative is d(u)/d(x)=−5

3. Rearrange the differential to solve for d(x) giving d(x)=−1/5*d(u)

4. Rewrite the integral in terms of u

(∫_^)(u6*(−1/5)*d(u))

5. Factor out the constant:

−1/5*(∫_^)(u6*d(u))

6. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)

−1/5*((u7)/7)+C

7. Simplify the expression:

−(u7)/35+C

8. Back-substitute u=4−5*x to return to the original variable:

−((4−5*x)7)/35+C

Final Answer

(∫_^)((4−5*x)6*d(x))=−((4−5*x)7)/35+C

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