2 (x 2 + a 2 ) x 2. is easy. Do this first.

Transcription

1 MAC 3 INTEGRATION BY PARTS General Remark: Unless specified otherwise, you will solve the following problems using integration by parts, combined, if necessary with simple substitutions We will not explicitly state domains on which these integrals make sense Exercise Let a 0 be a real number Find the integral x a Outline of Solution: i Write x a = x a a x a x x a ii Integrating a x a iii Now integrate is easy Do this first a = a x a x x a x x a Here, you will use integration by parts: choose a suitable u and dv iv Put the answers to ii and iii together, and you will get the following final answer: x a = x a arctan x 3 a a x a Exercise Find the integral arcsin x Answer: You will use integration by parts twice arcsin x = xarcsin x x arcsin x x Exercise 3 Find the integral x arcsin x x Answer: If you carefully choose u and dv, one integration by parts should do it x arcsin x x = x arcsin x x

2 Exercise 4 Find the integral arcsin x x 3 Outline of Solution: Use u = arcsin x and dv = x 3 The integral v du is going to look like x x This integral can be calculated by a trig substitution which we have not covered yet Rewrite the fraction as x x = x x x = x However, we can use integration by parts here x x x The first term is easy to integrate you get arcsin To integrate the second term, use integration by parts arcsin x = x x 3 x arcsin x x Exercise 5 Find the integral arcsin x arccos x Outline of Solution: i We will use the following trig formula: arcsin x arccos x = π/ You will first prove the formula You can do this, by setting α = arcsin x and β = arccos x Next, you recall that the domain of functions arcsin x and arccos x is the interval x Also, the range of α = arcsin x is π/ α π/ and the range of β = arccos x is 0 β π Using sine-of-sum rule from trig sinα β = sin αcos β cos αsin β, you should get sinα β = Why? This will show that that α β must equal π/ Why? ii Using the formula we can rewrite the integral as arcsin x π/ arcsin x = π/ arcsin x arcsin x

3 The second integral was found in Exercise, while the first one can be handled by a single integration by parts π x arcsin x x xarcsin x x arcsin x x Exercise 6 Find the integral x e arctan x x 3/ Answer: Let F x denote the answer for the integral If you choose your u and dv carefully, after integration by parts you will get an equation in F which you can solve for F x x earctan x Exercise 7 Find the integral x 7 x 4 Answer: Make a good choice for u and dv x 4 4 x 4 ln x4 4 Exercise 8 Let n be an integer, and let a and b be real numbers such that a b 0 Let I n denote the integral a sin x b cos x n Prove the following reduction formula: n a b Outline of Solution: Write n I n a sin x b cos x n = b sin x a cos x a sin x b cos x n a sin x b cos x n a sin x b cos x n Use integration by parts with u = a sin x b cos x n and dv = a sin x b cos x Eventually, you will end up with a formula that links I n and I n : b sin x a cos x I n = ni n a b n a sin x b cos x n 3

4 4 Now you can change n to n throughout the formula to get the needed reduction formula Exercise 9 Use the reduction formula from Exercise 8 to find the following integral: sin x cos x 3 Outline of Solution: After you do the reduction, the hardest part of your job will be to find sin x cos x This can be done as follows First, use trig to write Explain this carefully Then, it remains to integrate = 5 sinx arctan sin x cos x = 5 sinx arctan 5 sinx arctan = The inside cover of your book says that csc u du = ln csc u cot u Show that this answer is the same as csc u du = ln tanu/ 0 5 ln x tan arctan sin x cos x 0sin x cos x csc x arctan 5 Exercise 0 Let n be an integer, and let a and b be real numbers such that a = b Let I n denote the integral a b cos x n Prove the following reduction formula: b sin x n a b a b cos x n n 3a n a b I n n n a b I n

5 5 Outline of Solution: i Start with I n = a b cos x = a b cos x = n a b cos x n cos x = ai n b a b cos x n Use integration by parts in the last integral with u = and dv = cos x a b cos x n ii After you do integration by parts, the term b sin x will appear Rewrite this term with the help of the following identity: b sin x = a b aa b cos x a b cos x Verify this identity and use it to complete the proof of our reduction formula Exercise Let 0 < δ < be a real number Find the integral δ cos x Outline of Solution: Use the reduction formula of Exercise 0 with n =, a = and b = δ, which reduces the computation to finding δ cos x This integral was computed in Example 3 of handout Antiderivatives δ sin x δ δ cos x δ tanx/ arctan δ δ π x π, δ π for x π kπ, where k = 0, ±, ±, and denotes the greatest integer function For x = kπ π, we need to take the limit of the above function as x π kπ