Evaluate the Integral integral of (1-2x)^9 with respect to x

Problem

(∫_^)((1−2*x)9*d(x))

Solution

1. Identify the inner function for substitution, which is u=1−2*x

2. Calculate the differential d(u) by differentiating u with respect to x giving d(u)=−2*d(x)

3. Rearrange the differential to solve for d(x) resulting in d(x)=−1/2*d(u)

4. Substitute u and d(x) into the original integral to rewrite it in terms of u

(∫_^)(u9*(−1/2)*d(u))

5. Factor out the constant −1/2 from the integral.

−1/2*(∫_^)(u9*d(u))

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

−1/2*((u10)/10)+C

7. Simplify the expression by multiplying the fractions.

−(u10)/20+C

8. Back-substitute the original expression 1 - 2xƒ*o*r$ to get the final result.

Final Answer

(∫_^)((1−2*x)9*d(x))=−((1−2*x)10)/20+C

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