Evaluate the integral using integration by parts?

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SUMMARY

The integral in question, ∫ x f(x) dx from 0 to 1, requires the application of integration by parts, given the conditions f(1) = 6 and f'(1) = 7. The discussion reveals that the function f(x) was assumed to be x² + 5x, leading to confusion regarding the answer choices provided. The correct approach involves recognizing the need for integration by parts to derive the expressions for the integral, particularly focusing on the second derivative f''(x) as indicated in the answer choices. The participants concluded that further clarification on the integration by parts method is essential for solving the integral accurately.

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  • Knowledge of evaluating definite integrals
  • Basic proficiency in calculus concepts
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  • Review the integration by parts formula and its application in definite integrals
  • Study the properties of derivatives, particularly higher-order derivatives
  • Practice evaluating integrals involving polynomial functions
  • Explore examples of integrals that utilize integration by parts for clarification
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Homework Statement


Evaluate the integral.
Integral = x f(x) dx from 0 to 1 when f(1) = 6, f'(1) = 7.
Answer choices:
A. 11/6 + 1/6 integral from 0 to 1 x^3f''(x)dx
B. 11/12 - 1/6 integral from 0 to 1 x^3f''(x)dx
C. 11/3 + 1/2 integral from 0 to 1 x^2f'(x)dx
D. 11/3 - 1/2 integral from 0 to 1 x^2f'(x)dx
E. 11/4 - 1/2 integral from 0 to 1 x^2f"(x)dx

Homework Equations


The Attempt at a Solution


So just by looking at the answer choices, the method is integration by parts. Based on what was given, I determined that f(x)=x^2+5x. But that would give an integral of x(x^2+5x), which doesn't even need integration by parts to solve. I am stuck here and don't know how the answer choices are gotten.
 
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Did you try integrating by parts?
 

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