Integration By Substitution: Antiderivatives

Calculus: We introduce the technique of integration by substitution for indefinite integrals. This is an integration method that reverses the chain rule for derivatives.
Examples include (a) int x(1+x^2)^3 dx, (b) int x^6 cos(2x^7) dx, (c) int sin(x)cos(x) dx, and (d) int x(1+x)^2 dx. In problem (c), we recall an occasional problem with integrals, and, in problem (d), we show an instance where re-substitution is required.

Category:

Education

Tags:

• Mathematics
• calculus
• antiderivative
• integration
• substitution
• indefinite
• integral
• chain
• rule

[level2400]

Hey, how do you find a solution to an indefinite integral where the outside function is gone, or cannot match the derivative of the inside function, such as indefinite integral y^2(1 + y)^2, or indefinite integral (x^2 + 3)^2?

Can't find info on this ANYWHERE.

[MathDoctorBob]

@level2400 In both cases, your best bet is to expand into either y^4 +2y^3 + y^2 or x^4 + 6x^2 + 9 and use the power rule.

I understand what you mean though; we'll have special techniques for when the substitution is off, and these we consider on a case by case basis. These include logarithmic integrals, inverse trig substitutions, partial fractions, integration by parts, etc. - Bob

[Soriphen]

Awesome stuff! Took me a bunch of searching to finally find the video that explains every confusion I've had with integration by substitution. Very thankful for the amount of time you put into these vids :D.

[MathDoctorBob]

@Soriphen You're welcome, and thanks! Glad to be of help. - Bob

[MrAverypfeiffer]

Dr. Bob you are a life saver. i have an AP calculus AB test tommorrow that i just found out about today. I had missed the previous two days out sick. Apparently during those days my teacher decided to teach integration through substitution. if it wasnt for you i would be failing that test tommorrow. again thank you! i intend to use your videos more often. keep up the good work!

[MathDoctorBob]

@MrAverypfeiffer Thanks for the kind words, and good luck with the exam! If there are any problems that you want to see, please let me know. - Bob

[level2400]

@MathDoctorBob Thanks, that makes complete sense. Love the vids! Getting me through Calc2.

[MathDoctorBob]

@limpbits  You're welcome! I hear you. Math is the gatekeeper. - Bob

[limpbits]

swear my upper division chemistry classes are cake compared to cal2, and I instinctively know if i can just make it to differential eq. i can ace any subject. thanks for clarifying this stuff.

[MathDoctorBob]

@sophia91chan You need to identify a composition in the integral. g(x) will be the inside function; if you evaluate at a number, g(x) is the first function that you compute. - Bob