MIT grad shows how to integrate using trigonometric substitution. To skip ahead: 1) For HOW TO KNOW WHICH trig substitution to use (sin, tan, or sec), skip to 0:18. 2) For how to do a SIN SUB and WHEN TO TRY TRIG SUB, skip to 1:29. 3) For a TAN SUB with the radical in the NUMERATOR, skip to 14:41. 4) For a SEC SUB with a RATIONAL/FRACTIONAL POWER, skip to 22:36. 5) For miscellaneous types, skip to time 24:54. Trig substitution integration is a calculus technique for integrals. Nancy formerly of MathBFF explains the steps.
For my video on how to do BASIC INTEGRATION, jump to: https://youtu.be/e1nxhJQyLYI
For my video on how to integrate using U-SUBSTITUTION, jump to: https://youtu.be/8B31SAk1nD8
For my video on how to do INTEGRATION BY PARTS, jump to: https://youtu.be/KKg88oSUv0o
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HOW TO KNOW WHICH trig substitution to use: If the x-expression under the radical has the form of "a number minus an x^2 term", then you'll use a SINE substitution. If it has the reversed form of "an x^2 term minus a number", then you'll use a SECANT substitution. And finally, if the expression under the root is addition (the form of "an x^2 term plus a number"), you'll use a TANGENT substitution for the problem.
WHEN to use TRIG SUB: If you have an integral with a radical expression in it, and you know you cannot integrate using: a Table of Integrals integration rule, or using U-Substitution, or using the Power Rule, or using other integration techniques like Integration by Parts or Partial Fractions, then it is a good idea to try trig substitution.
HOW TO DO TRIG SUB. Here are the main steps of trig sub problems: 1) Decide which trig substitution you need [x = asin(theta), x = atan(theta), or x = asec(theta)], filling in the value for "a". 2) Simplify the radical expression by plugging in your substitution for x in the radical [For example, for a sin sub, plug in for x in sqrt(a^2-x^2)]. 3) Find dx from your x substitution. 4) Substitute into the original integral, replacing the radical, the dx, and any other x factors, with the theta expressions you found. 5) Simplify and integrate, using whatever technique you have to use. 6) After you integrate, it's still in terms of theta. So the last step is to get it all back in terms of x. Draw a right triangle off to the side, label the sides with what you know from your trig substitution expression, find the third side, and use the sides of the triangle to get an expression for whatever trig function you need in your final answer.
For more of my calculus antiderivative, indefinite integral, definite integral, and derivative math videos, as well as more examples and help with integration rules, using an integral table, integration formulas, trig integrals, completing the square, and a review of right triangle trigonometry, check out: http://nancypi.com