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## Integral Calculus

### Course: Integral Calculus>Unit 1

Lesson 17: Integration by parts

# Integration by parts: ∫x²⋅𝑒ˣdx

Integration by parts helps find antiderivatives of products of functions. We assign f(x) and g'(x) to parts of the product. Then, we find f'(x) and g(x). The formula is ∫f(x)g'(x)dx = f(x)g(x) - ∫f'(x)g(x)dx. Sometimes, we use integration by parts twice! Created by Sal Khan.

## Want to join the conversation?

• At the last step , could we also express this as e^x(x^2-2x+2)+c ?
ty • At he takes the anti-derivative of e^x. Wouldn't the anti-derivative be e^x + c?

If that is the case, does that mean you would multiply the constant by 2 if you substitute it back in? •   This is a point that many students get hung up on. Does multiplying the arbitrary constant change that constant? Sal has mentioned many times in other videos that this is not the case.
Multiplying an arbitrary constant by some number just yields another arbitrary constant. If you had kept with the convention of 2*c, you would solve for c (if you had an initial condition) and plugged it in. If I had replaced 2*c with another constant, c_0, my value for c_0 would just end up being twice your value. Personally, I do away with the c_0 because an arbitrary constant is just an arbitrary constant.
Another point, performing any operation on an arbitrary constant just yields another arbitrary constant. Addition, subtraction, exponentiation...
• So what happens if someone does the wrong assignment to f(x) and g'(x) from the beginning ?? I tried it from myself and I got a pretty wacky result : e^x(x^5/60 - x^4/12 + x^3/3). Is it true that if you do the wrong assignment you'll get the wrong result? •  yep, most of the time you end up wasting your time getting nowhere.
Use LIPET to choose the u part of the substitution. (the other factor is the dv)
Logarithms
Inverse Trigs
Polynomials
Exponentials
Trig
TRUST in LIPET it will save many a headache from happening.
It's right up there with PEMDAS and SOHCAHTOA
• what would you do if e^x had a number and a power with it e.g. 5e^3x? • Stupid question: Why can't I just make it like this:
Antiderivative of x^2 = x^3/3
Antiderivative of e^x = e^x
Therefore the antiderivative of x^2*e^x is x^3/3*e^x • Hi, how do i solve for the integral of e^(x^2)dx? I've been stuck for some time now. • That is an advanced-level integral: it cannot be solved by ordinary integration methods taught in an introductory integral calculus course. The answer involves non-elementary functions, specifically it involves the imaginary error function.

I have only a passing knowledge of this, so I will leave it to a professional mathematician to show you how to reach the answer. But, I thought it good to let you know this problem cannot be solved by ordinary integration methods.
• Integration by parts of (x^2)*(e^-x^2/2) where by the upper limit of the integral is infinity and lower limit is 0 • This was a struggle but I think I might have got it.

I used integration by parts and separated the integral to
x * (x * e^-x^2/2)

Then I got stuck when that creates the integral of e^-x^2/2
but apparently this is almost equal to "erf" (something I never heard of before!)
erf(x) = 2/sprt(pi) integral(e^-t^2)

By making that substitution I got the result of the integral:
-xe^-x^2/2 + sqrt(2)/2*pi*erf(sqrt(2)/2 * x)

Evaluating the bounds the first part -xe^-x^2/2
evaluates to 0 for both 0 and infiniti so ignore it.

Then I learned that
erf(0) = 0
erf(infiniti) = 1

sqrt(2)/2 * pi
(1 vote)
• Why is the anti-derivative of x^2 2x? I thought that it would be (x^3)/3 because of the form (x^n+1)/n+1. • when he takes the two out front of the integral at , do you have to do that to get the correct answer or does hat just make things easier? • Just a random thought. The symbol of integration has a shape of squeezed in width, and stretched in height S since it stands for sum, doesn't it? And the "dx" at the end indicates that we're multiplying the infinite amount of rectangles which approximate the height of the curve at the respective points by the infinitely small change in our independent variable, which represents the base of these rectangles, right? What is the point of evaluating a sum of the rectangles multiplied by their base in no specific interval? And why calling that sum as an anti-derivative? It doesn't make any sense to me ... . The anti-derivative is a function, which has nothing to do with the sum. I understand it might be a convention, but ... isn't it quite a misleading one? • You are not "evaluating a sum of the rectangles multiplied by their base" the base is dx. The section on Riemann sums and taking their limit as Δx→0 covers all this. When Δx is infinitesimal, it is called dx.

The fundamental theorem of calculus shows you why it is called the anti-derivative. The anti-derivative is the inverse of the derivative. Say you integrate a function f, and call that result F. If you then take the derivative of F you get back f. The FToC says it more formally this way:

Given a continuous function, f, on an open interval I with α being any point in I,
IF you define F as:
F = ∫f(t)dt from α to x
THEN
F'(x) = f(x)

It doesn't get much clearer or obvious than that as to the relationship between the derivative and anti-derivative (that is, that the derivative of the anti-derivative of f gets you back your original f).