Evaluate the Integral

Problem

(∫_^)(x9*e(x10)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x10 is a multiple of the other factor x9

2. Substitute u=x10 to simplify the integrand.

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

d(u)/d(x)=10*x9

4. Rearrange the differential to solve for the terms present in the integral.

1/10*d(u)=x9*d(x)

5. Rewrite the integral in terms of u

(∫_^)(eu⋅1/10*d(u))

6. Factor out the constant from the integral.

1/10*(∫_^)(eu*d(u))

7. Integrate the exponential function.

1/10*eu+C

8. Back-substitute x10 for u to return to the original variable.

1/10*e(x10)+C

Final Answer

(∫_^)(x9*e(x10)*d(x))=(e(x10))/10+C

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