# Math 19B Midterm Review - Math 19B Midterm 2 Study Aid...

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Math 19B Midterm 2 Study Aid November 16, 2010 Pre-Chapter 7.1 - Stu ff you Should Know! Basic Antiderivatives. The following is the minimum list of antiderivatives you should have either memorized or written on your review card ( C and c are constants): c dx = cx + C x n dx = x n +1 n + 1 + C ( n = 1) 1 x dx = ln | x | + C e x dx = e x + C sin x dx = cos x + C cos x dx = sin x + C sec 2 x dx = tan x + C f ( x ) f ( x ) dx = ln | f ( x ) | + C 1 x 2 + a 2 dx = 1 a arctan x a + C 1 1 x 2 dx = arcsin x + C U-substitution : For indefinite integrals the u -subs formula may be expressed: f ( g ( x )) g ( x ) dx = f ( u ) du where u = g ( x ). In the definite case the formula reads: b a f ( g ( x )) g ( x ) dx = g ( b ) g ( a ) f ( u ) du. Chapter 7.1 - Integration by parts. Formula. Integration by parts formula. f ( x ) g ( x ) dx = f ( x ) g ( x ) g ( x ) f ( x ) dx or, in the more catchy form: u dv = uv v du. 1
Remarks. ILATE. When considering the product of two functions in an integration by parts problem, we need to choose one function to be u and one to be dv . For your ‘first attempt’ choose u by ‘order of appearance’ in the ‘ILATE’ sequence: I I nverses (trig inverses such as arcsin x etc.) L L ogarithms (usually ln x ) A A polynomial (either x , x 2 or x 3 .) T T rig function (sin( x ), cos(2 x ), etc.) E = E xponential ( e x , e 3 x , etc.). Multiply by 1. Remember you can write f ( x ) as 1 × f ( x ). This trick is commonly used in the ‘ I ’ or L ’ cases of ILATE; notably in calculating ln x dx or arcsin x dx. Multiple integration by parts. If you choose u = x 2 in the ‘ A ’ case of ILATE you may have to do integration by parts (IBP) twice. Similarly, if you’re forced to choose u = x 3 , you may have to do IBP three times! In the ‘ T ’ or ‘ E ’ cases of ILATE you’ll probably have do IBPs twice and use the trick that the integral you get the second time around is exactly what you started with. If you haven’t seen this type of example or have no idea what I’m going on about (!) try: e 4 x sin(2 x ) dx. Chapter 7.2 - Trigonometric integrals. Formula. Here are all the trig identities you should need: sin 2 x + cos 2 x = 1 sin 2 x = 1 2 (1 cos(2 x )) tan 2 x + 1 = sec 2 x cos 2 x = 1 2 (1 + cos(2 x )) 1 + cot 2 x = csc 2 x sin(2 x ) = 2 sin x cos x. Techniques. sin m x cos n x dx . 1. If m is odd, factor out one sin x and use sin 2 + cos 2 = 1 to express the rest of the integral in terms of cos x . Then make the u-subs, u = cos x . 2. If n is odd, factor out one cos x and use sin 2 + cos 2 = 1 to express the rest of the integral in terms of sin x . Then make the u-subs, u = sin x . 3. If m and n are both even use the half angle (‘power reducing’) identities: sin 2 x = 1 2 (1 cos(2 x )) and cos 2 x = 1 2 (1 + cos(2 x )) . tan m x sec n x dx . Page 2
1. If n is even (think s E c = E ven), factor out one sec 2 x and use tan 2 +1 = sec 2 x to express the rest of the integral in terms of tan x . Then make the u-subs, u = tan x .