Evaluate the integral using integration by parts?

That would be the next step. In summary, the integral can be evaluated using the method of integration by parts and the given information about f(x) and its derivatives. The correct answer choice is not immediately obvious and further computation is required to determine the integral.
  • #1
turbokaz
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0

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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  • #2
Did you try integrating by parts?
 

Related to Evaluate the integral using integration by parts?

1. What is integration by parts?

Integration by parts is a method used in calculus to evaluate integrals of products of functions. It involves breaking down a complex integral into simpler ones, and then using a specific formula to solve for the final result.

2. When should I use integration by parts?

Integration by parts is useful when the integral involves a product of functions, one of which can be easily integrated while the other remains in its original form. This method is also helpful when dealing with logarithmic, trigonometric, or exponential functions.

3. What is the formula for integration by parts?

The formula for integration by parts is ∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx, where u(x) and v(x) are two functions and u'(x) and v'(x) are their derivatives.

4. How do I choose which function to use as u(x) and v'(x)?

When using integration by parts, it is important to choose u(x) and v'(x) in a way that simplifies the integral. A general rule of thumb is to select u(x) as the function that becomes simpler when differentiated, and v'(x) as the function that becomes easier to integrate when differentiated.

5. Are there any limitations to using integration by parts?

Integration by parts can only be used when the integral can be expressed as a product of two functions. It is also not effective when the integral contains more than one variable or when the functions involved are too complex to integrate. In such cases, other integration techniques may be more suitable.

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