Study guides › 9231 FP2 › Integration Maths4U Original · Learn, check, explain
Integration techniques — 9231 FP2
Choose substitution, inverse forms, partial fractions or integration by parts.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods A substitution must transform the whole integrand, differential and limits. Recognise \(\displaystyle \int \frac{x}{x^{2}+ a^{2}}\, dx= \frac{1}{2}\ln (x^{2}+ a^{2})+ C\), and \(\displaystyle \int \frac{dx}{\sqrt{x^{2}+ a^{2}}}= \operatorname{arsinh}(\frac{x}{a})+ C\) for a>0.
Use integration by parts \(\int u\,dv=uv-\int v\,du\) when differentiating one factor simplifies it. A proper rational function may instead be decomposed into partial fractions. Check an antiderivative by differentiation.
Check the conditions Transform definite limits when changing variables. Retain absolute values in logarithms where the domain needs them. Three worked examples Example 1 · Advanced integration techniques Evaluate the following independent definite integrals on [0,1]. Show a suitable method.
Evaluate \(\displaystyle \int_{0}^{1} \frac{x}{x^{2}+ 9}\, dx\). Evaluate \(\displaystyle \int_{0}^{1} (x^{2}+ 9)^{- \frac{1}{2}}\, dx\). Evaluate \(\int_0^1 xe^{2x}\,dx\). Evaluate \(\displaystyle \int_{0}^{1} \frac{1}{[(x+ 3)(x+ 4)]}\, dx\). Show the reasoning (a) \(\displaystyle \frac12[\ln(x^2+9)]_0^1=0.0526802578289132\)
Answer: \(0.0526802578289132\).
(b) \(\displaystyle [\operatorname{arsinh}(\frac{x}{3})]_{0}^{1}= 0.327450150237258\)
Answer: \(0.327450150237258\).
(c) \(\displaystyle [e^{2x}(\frac{x}{2}- \frac{1}{4})]_{0}^{1}= 2.09726402473266\)
Answer: \(2.09726402473266\).
(d) \([\ln(x+3)-\ln(x+4)]_0^1=0.0645385211375712\)
Answer: \(0.0645385211375712\).
Example 2 · Advanced integration techniques Evaluate the following independent definite integrals on [0,1]. Show a suitable method.
Evaluate \(\displaystyle \int_{0}^{1} \frac{x}{x^{2}+ 16}\, dx\). Evaluate \(\displaystyle \int_{0}^{1} (x^{2}+ 16)^{- \frac{1}{2}}\, dx\). Evaluate \(\int_0^1 xe^{3x}\,dx\). Evaluate \(\displaystyle \int_{0}^{1} \frac{1}{[(x+ 4)(x+ 5)]}\, dx\). Show the reasoning (a) \(\displaystyle \frac12[\ln(x^2+16)]_0^1=0.0303123109082174\)
Answer: \(0.0303123109082174\).
(b) \(\displaystyle [\operatorname{arsinh}(\frac{x}{4})]_{0}^{1}= 0.247466461547263\)
Answer: \(0.247466461547263\).
(c) \(\displaystyle [e^{3x}(\frac{x}{3}- \frac{1}{9})]_{0}^{1}= 4.57456376070837\)
Answer: \(4.57456376070837\).
(d) \([\ln(x+4)-\ln(x+5)]_0^1=0.0408219945202552\)
Answer: \(0.0408219945202552\).
Example 3 · Advanced integration techniques Evaluate the following independent definite integrals on [0,1]. Show a suitable method.
Evaluate \(\displaystyle \int_{0}^{1} \frac{x}{x^{2}+ 25}\, dx\). Evaluate \(\displaystyle \int_{0}^{1} (x^{2}+ 25)^{- \frac{1}{2}}\, dx\). Evaluate \(\int_0^1 xe^{4x}\,dx\). Evaluate \(\displaystyle \int_{0}^{1} \frac{1}{[(x+ 5)(x+ 6)]}\, dx\). Show the reasoning (a) \(\displaystyle \frac12[\ln(x^2+25)]_0^1=0.0196103565766407\)
Answer: \(0.0196103565766407\).
(b) \(\displaystyle [\operatorname{arsinh}(\frac{x}{5})]_{0}^{1}= 0.198690110349241\)
Answer: \(0.198690110349241\).
(c) \(\displaystyle [e^{4x}(\frac{x}{4}- \frac{1}{16})]_{0}^{1}= 10.2996531312145\)
Answer: \(10.2996531312145\).
(d) \([\ln(x+5)-\ln(x+6)]_0^1=0.0281708769666962\)
Answer: \(0.0281708769666962\).
Common mistakes Do not mix old limits with a new variable. The subtracted integral in integration by parts has a minus sign. Practise with answer checking Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
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