Integration by Parts: Solving ∫x*e^-x dx

In summary, Integration by Parts is a technique used in calculus to evaluate integrals that are in the form of the product of two functions. Its formula is ∫u*dv = uv - ∫v*du, and it involves choosing u and dv based on which function becomes simpler when differentiated and which becomes easier to integrate. The purpose of Integration by Parts is to simplify complex integrals and make them easier to solve. To solve ∫x*e^-x dx, you would choose u = x and dv = e^-x and use the formula ∫u*dv = uv - ∫v*du to get the final answer of -x*e^-x + e^-x + C.
  • #1
p.mather
19
0

Homework Statement



∫ x * e^-x dx

Homework Equations



Integration by parts: Just wondering if below is correct. Not brilliant with Integration by parts and not sure if my +ve and -ve signs are correct. Some help to say if i am correct or where i have gone wrong would be brilliant.

The Attempt at a Solution



= x * ((e^-x)/(-1)) -∫ ((e^-x)/(-1)) * 1 dx

= -x * e^-x + ∫ e^-x dx

= -x * e^-x + ((e^-x)/(-1)) +c

= -x * e^-x - e^-x +c
 
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  • #2
Yes.. That is right.. If u are not certain just differentiate your ans..
 

What is Integration by Parts?

Integration by Parts is a technique used in calculus to evaluate integrals that are in the form of the product of two functions. It involves breaking down the integral into smaller parts and using the product rule of differentiation to solve it.

What is the formula for Integration by Parts?

The formula for Integration by Parts is ∫u*dv = uv - ∫v*du, where u and v are the two functions in the integral and du and dv are their respective derivatives.

How do I know which function to choose for u and dv?

The general rule of thumb is to choose u as the function that becomes simpler when differentiated, and dv as the function that becomes easier to integrate. However, in some cases, trial and error may be required to determine the best choice for u and dv.

What is the purpose of Integration by Parts?

The purpose of Integration by Parts is to simplify complex integrals and make them easier to solve. It can also be used to solve integrals that are not solvable by other methods.

How do I solve ∫x*e^-x dx using Integration by Parts?

To solve ∫x*e^-x dx, you would choose u = x and dv = e^-x. Then, you would use the formula ∫u*dv = uv - ∫v*du to solve the integral. In this case, u would become simpler when differentiated and dv can be easily integrated to give the final answer of -x*e^-x + e^-x + C.

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