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Math2org Math Tables Integral ln (x) Discussion of ln (x) dx = x ln (x) x C 1 Proof Strategy Use Integration by Parts ln (x) dx set u = ln (x), dv = dxShop the latest women's fashion at PrettyLittleThing Hit refresh on your wardrobe and discover thousands of this season's musthave looks and trends online at PLTSolve the integral = ln u C substitute back u=cos x = ln cos x C QED 2 Alternate Form of Result tan x dx = ln cos x C = ln (cos x)1 C = ln sec x C

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U = ln(x), dv = dx then we find du = (1/x) dx, v = x substitute ln(x) dx = u dv and use integration by parts = uv v du substitute u=ln(x), v=x, and du=(1/x)dx = ln(x) x x (1/x) dx = ln(x) x dx = ln(x) x x C = x ln(x) x C QEDIntegrate ln(u) = 3 ln(x) C Make C = −ln(k) ln(u) ln(k) = 3ln(x) Then uk = x 3 And so u = x 3 k Step 5 Substitute u back into the equation at Step 2 (Remember v term equals 0 so can be ignored) ( x 3 k) dv dx = x Step 6 Solve this to find v Separate variables dv = k x2 dx Put integral sign ∫ dv = ∫ k x2 dx Integrate v = −k x1 D Step 7 Substitute into y = uvDefinition Let u = a i b j c k and v = d i e j f k be vectors then we define the cross product v x w by the determinant of the matrix We can compute this determinant as = (bf ce) i (cd af) j (ae bd) k Example Find the cross product u x v if u = 2 i j 3 k v = 4 j 5 k Solution

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Yes, yn = E xk is causal because the value of y at any instant n depends only on the previous (past) values of x Invertible (b) y(t) = x(2t) is invertible;Use Substitution sec x dx = sec x sec x tan x sec x tan x dx set u = sec x tan xX(t) = y(t/2) (c) yn = E _xk is not invertible Summation is not generally an invertible operation

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 h X^ X ( u µ v o í U ó î ò U ì ñ ï µ v U Ç r } À r Ç v } ( ô ì X î 9 ( } u D Ç î ì í õ X > l o Ç v P o Z v P µ v= e u C = e x 2 2x3 C Of course, it is the same answer that we got before, using the chain rule "backwards" In essence, the method of usubstitution is a way to recognize the antiderivative of a chain rule derivative Here is another illustraion of usubstitution Consider Let u = x 3 3x Then (Go directly to the du part) du = (3x 2 3) dx = 3(x 2 1) dx, so that (1/3) du = (x 2 1C Ó X U j y ò ë " ü U 0 í Õ !

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í Õ C f d à d X U p Ï U È ò ë " ü U V Ã Ó Õ f ý Ó X U C ½ Ô 9 ì Õ Ó X U U Ó U !Proof Strategy The strategy is not obvious Multiply and divide by (sec x tan x);Set u = sec x tan x then we find du = (sec x tan x sec 2 x) dx substitute du = (sec x tan x sec 2 x) dx, u = sec x tan x

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Z z z f h q wu d od y h q x h f k u \ v oh u mh h s f r p h h s wk h x q g lv s x wh g lq j r i wk h r ii u r d g d g y h q wx u h lq y lwh v \ r x wr f olp e lq wr wk h g u ly h uWe use cookies that help us provide you with the best possible shopping experience with us For example, they allow us to connect to social networks, display personalised content, as well as analyse and improve the operation of our website Accept All More Info Update your cookie settings We use strictly necessary cookies You can't disable these They allow features likeX = how far along m = Slope or Gradient (how steep the line is) b = value of y when x=0 How do you find "m" and "b"?

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