Evaluate the Integral

Problem

(∫_0^1)(x^11*(1+x^12)^11*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of the inner function 1+x^12 is proportional to the outer factor x^11

2. Define the substitution variable u=1+x^12

3. Differentiate u with respect to x to find d(u)=12*x^11*d(x) which implies x^11*d(x)=1/12*d(u)

4. Change the limits of integration from x to u when x=0 u=1+0^12=1 when x=1 u=1+1^12=2

5. Substitute these values into the integral to rewrite it in terms of u

(∫_1^2)(u^11⋅1/12*d(u))

6. Factor out the constant and apply the power rule for integration.

1/12*(∫_1^2)(u^11*d(u))

1/12*[(u^12)/12]^2_1

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

1/144*(2^12−1^12)

1/144*(4096−1)

4095/144

8. Simplify the resulting fraction by dividing both the numerator and denominator by their greatest common divisor, 9.

(4095÷9)/(144÷9)=455/16

Final Answer

(∫_0^1)(x^11*(1+x^12)^11*d(x))=455/16

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