REVERSE CHAIN RULE CALCULUS 7. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Reverse Chain Rule 1/12 Adrian Jannetta

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1 REVERSE CHAIN RULE CALCULUS 7 INU05/55 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Reverse Chain Rule /2 Adrian Jannetta

2 Reversing the chain rule In differentiation the chain rule is used to get the derivative of a composite function. For example d dx (e2x )=2e 2x Integration is the reverse of differentiation. Therefore, we could reverse this statement and write: 2e 2x dx= e 2x (Ignore the constant for a moment). Dividing both sides by 2 we find e 2x dx= e2x 2 () Similarly, consider d dx [sin(3x+2)]=3cos(3x+2). Reversing this process: 3cos(3x+2)dx=sin(3x+2) Therefore: cos(3x+2)dx= sin(3x+2) 3 Can you a pattern for directly integrating the expressions given in () and (2)? Reverse Chain Rule 2/2 Adrian Jannetta (2)

3 Reversing the chain rule Here s another example of reversing the chain rule. d dx (2x 5)3 =(3)(2)(2x 5) 2 In words this process is Subtract from the power. Multiply by the old power and multiply by the derivative of the inner function. Since integration is the reverse of differentiation: (3)(2)(2x 5) 2 dx=(2x 5) 3 (Again, let s ignore the constant C for a moment). Dividing both sides by the factor(3)(2) we get (2x 5) 2 dx= (2x 5)3 (3)(2) Notice that the (3) is the power and the (2) is the derivative of the inner function(2x 5) (2x 5) 2 dx= (2x 5)3 6 The first term on the RHS is equivalent in words to Add to the power. Divide by the new power and divide by the derivative of the inner function. It is the reverse of applying the chain rule. Reverse Chain Rule 3/2 Adrian Jannetta

4 Reverse Chain Rule The examples just seen illustrate a simple procedure for integrating composite functions. It is the reverse of how we d carry out the chain rule in differentiation. In words, the general procedure doing reverse chain rule is Integrate the outer function. Divide by the derivative of the inner function Here some examples. We ll include the constant from now on! sin0x dx= cos0x + C 0 e 4x dx= e 4x 4 + C Reverse Chain Rule 4/2 Adrian Jannetta

5 Reverse chain rule Use the reverse chain rule to integrate First, write the integral in a more suitable form (3 5x) 4 dx (3 5x) 4 dx Now integrate. Use the rule for integrating powers (in this case, like x 4 ); Add to the power. Divide by the new power and divide by the derivative of the inner function. (3 5x) 4 dx= (3 5x) 3 ( 3)( 5) + C=(3 5x) 3 + C 5 It can be written without the negative power like this: (3 5x) dx= 4 5(3 5x) 3+ C Reverse Chain Rule 5/2 Adrian Jannetta

6 When can reverse chain rule be used? The reverse chain rule can only be used when the inner function is linear, which means it has the form mx + c (where m and c are constants). We cannot use the reverse chain rule when the inner function is NOT LINEAR! For example, we could not use it to find function +x 2 is not linear. Therefore (+x 2 ) 5 dx (+x2 ) 6 + C 6(2x) (+x 2 ) 5 dx because the inner (You can check this by finding the derivative of the RHS - it will not be equal to the LHS!) Reverse Chain Rule 6/2 Adrian Jannetta

7 Test yourself Which of the following can be integrated with the reverse chain rule? ( 3x) 4 dx 2e x dx x 2 dx 3x dx sinx dx cos(3x π) dx +2x dx 2x sinx dx Answers... Yes 3 No 5 No 7 Yes 2 Yes 4 Yes 6 Yes 8 No Reverse Chain Rule 7/2 Adrian Jannetta

8 Further examples Rational power Find 2x+5dx with the reverse chain rule. First rewrite without the square-root: (2x+5) 2 dx = (2x+5) 3 2 ( 3 2 )(2) + C = (2x+5) C 3 (2x+5) 3 = + C 3 You can always check the answer by differentiating it! You should end up with the original integrand. Reverse Chain Rule 8/2 Adrian Jannetta

9 A definite integral Evaluate 0 (2x+) 3 dx Using the reverse chain rule we see that (2x+) 3 dx = Looking forward... 0 = 8 = 8 (2x+) 4 (4)(2) 0 (2x+) = 8 (8 ) = 0 The reverse chain rule is a useful shortcut for simple situations. A related technique, called integration by substitution, will be demonstrated in the next lecture. Reverse Chain Rule 9/2 Adrian Jannetta 0

10 Integration of f (x)/f(x) Here is a result that we saw during our study of differentiation: d dx ln f(x) = f (x) f(x) It s a consequence of differentiating logarithm functions using the chain rule. Naturally, we can reverse this to produce an integration result: f (x) f(x) dx=ln f(x) +C (3) This is a result we should learn to recognise when it appears! Reverse Chain Rule 0/2 Adrian Jannetta

11 Integration of f (x)/f(x) 2x Find x 2 9 dx. This integral matches the form of equation (3) exactly. Therefore: Integration of f (x)/f(x) 4sin x Find +cos x dx. 2x x 2 9 dx= ln x C This integral almost fits the form of equation (3). We can rewrite it to make it the same: 4sin x sin x +cosx dx= 4 +cos x dx We took out a factor of 4 so that the numerator is now the derivative of the denominator. Therefore: 4sin x dx= 4ln +cos x +C +cosx Reverse Chain Rule /2 Adrian Jannetta

12 Test yourself Use of reverse chain rule Use the reverse chain rule to find these integrals: ( 3x) 4 dx 5 π 9 0 sin 3x dx e x dx 5+2x dx 2cos( 3 x π) dx x dx cosx sin x dx Answers... 5 ( 3x)5 + C 2 2e x + C 3 2 ln 5+2x +C 4 6sin( 3 x π)+c ln sin x + C Reverse Chain Rule 2/2 Adrian Jannetta