Evaluate the Integral

Problem

(∫_0^1)(x9*(1+x10)9*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of x10 is related to the x9 term.

2. Substitute u=1+x10 which implies that the differential d(u)=10*x9*d(x)

3. Rearrange the differential to solve for the terms in the integral: 1/10*d(u)=x9*d(x)

4. Change the limits of integration according to the substitution: when x=0 u=1+0=1 when x=1 u=1+1=2

5. Rewrite the integral in terms of u

(∫_1^2)(1/10*u9*d(u))

6. Integrate with respect to u using the power rule:

1/10*[(u10)/10]21

7. Evaluate the expression at the upper and lower limits:

1/100*(2−1)

8. Simplify the numerical values:

1/100*(1024−1)

1023/100

Final Answer

(∫_0^1)(x9*(1+x10)9*d(x))=10.23

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