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A LevelAQAEdexcelOCR A Level The derivative of e^x is e^x. It follows that:int e^xdx=e^x+cDont forget +c.Indeed, more generally:int e^ax+bdx=dfrac1ae^ax+b+cExample: The integral of e^3x+4 is dfrac13e^3x+4 + cA Level Generally, int x^ndx=dfrac1n+1x^n+1+c. However, when n=-1, this involves dividing by 0. So another function to integrate to must be found. This is the natural logarithm.LARGE intdfrac1xdx=ln(|x|)+cThis rule also comes in two other forms:LARGE intdfrac1ax+bdx=dfrac1aln(|ax+b|)+cLARGE intdfracf(x)f(x)dx=ln(|f(x)|)+cExamples: LARGE intdfrac16-7xdx=-dfrac17ln(|6-7x|)+cLARGE intdfrac3cos(3x)sin(3x)dx=ln(|sin(3x)|)+c because the derivative of sin(3x) is 3cos(3x) so it is of the form LARGE intdfracf(x)f(x)dxA Level To integrate partial fractions, split them into multiple terms then apply the appropriate integration rule to each term.Example: Integrate dfrac5x+4(2x+1)(x+2) Step 1: split it into partial fractions.dfrac5x+4(2x+1)(x+2)=dfracA2x+1+dfracBx+25x+4=A(x+2)+B(2x+1)5x+4=Ax+2A+2Bx+BA+2B=52A+B=42A+4B=103B=6B=22A+2=42A=2A=1dfrac5x+4(2x+1)(x+2)=dfrac12x+1+dfrac2x+2 Step 2: Put it into the integral.beginalignedintdfrac5x+4(2x+1)(x+2)dx&=intdfrac12x+1+dfrac2x+2dx1.2em&=intdfrac12x+1dx+2intdfrac1x+2dxendaligned Step 3: Apply the rule to each term:LARGE intdfrac12x+1dx=dfrac12ln(|2x+1|)+cLARGE intdfrac1x+2dx=ln(|x+2|)+c Step 4: Put it all together.LARGE intdfrac5x+4(2x+1)(x+2)dx=dfrac12ln(|2x+1|)+2ln(|x+2|)+c A Level Example Questionsi)beginalignedint e^2xdx=dfrac12e^2x+cendaligned ii)beginalignedint left( 3e^5x+1right) dx&=left( 3timesdfrac15e^5x+1right) +c1.2em&=dfrac35e^5x+1+cendaligned iii)beginalignedint12e^9-28xdx&=left( 12times -dfrac-28e^9-28xright) +c1.2em&=-dfrac1228e^9-28x+c1.2em&=-dfrac37e^9-28x+cendalignedi)beginalignedintdfrac13x+1dx=dfrac13ln(|3x+1|)+cendaligned ii)beginalignedintdfrac3x+4dx&=left( 3timesintdfrac1x+4right) +c1.2em&=3ln(|x+4|)+cendaligned iii)beginalignedintdfrac-26-11xdx&=intdfrac211x-6dx1.2em&=2timesintdfrac111x-6dx1.2em&=left( 2timesdfrac111ln(|11x-6|)right) +c1.2em&=dfrac211ln(|11x-6|)+cendalignedi)beginalignedintdfrac6x+53x^2+5x+4dx&=intdfracdfracddx(3x^2+5x+4)3x^2+5x+4dx1.2em&=ln(|3x^2+5x+4|)+cendaligned ii)beginalignedintdfracx^3x^4+5dx&=dfrac14intdfrac4x^3x^4+5dx1.2em&=dfrac14intdfracdfracddx(x^4+5)x^4+5dx1.2em&=dfrac14ln(|x^4+5|)+cendaligned iii)beginaligned&intdfraccos(2x)sin(x)cos(x)dx=intdfraccos^2(x)-sin^2(x)sin(x)cos(x)dx1.2em&=intdfraccos(x)dfracddx(sin(x))+sin(x)dfracddx(cos(x))sin(x)cos(x)dx1.2em&=intdfracdfracddx(sin(x)cos(x))sin(x)cos(x)dx1.2em&=ln(|sin(x)cos(x)|)+cendalignedSplit into partial fractions: beginaligneddfrac11x+73x^2+7x+2&=dfrac11x+7(3x+1)(x+2)1.2em&=dfracA3x+1+dfracBx+2endaligned 11x+7=A(x+2)+B(3x+1) 11x+7=Ax+2A+3Bx+B A+3B=11 2A+B=7 2A+6B=22 5B=15 B=3 2A+3=7 2A=4 A=2 dfrac11x+73x^2+7x+2=dfrac23x+1+dfrac3x+2 Now integrate: beginalignedintdfrac11x+73x^2+7x+2dx&=intleft( dfrac23x+1+dfrac3x+2right) dx1.2em&=dfrac23ln(|3x+1|)+3ln(|x+2|)+cendaligned Related Topics Additional Resources Worksheet and Example QuestionsYou May Also Like... The MME A level maths predicted papers are an excellent way to practise, using authentic exam style questions that are unique to our papers. Our examiners have studied A level maths past papers to develop predicted A level maths exam questions in an authentic exam format. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country. From: £10.99 £9.99View Product A level maths revision cards and exam papers for Edexcel. MME is here to help you study from home with our revision cards and practice papers. The profit from every bundle is reinvested into making free content on MME, which benefits millions of learners across the country. From: £24.98 £22.99View Product The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country. £14.99View Product Online exams, practice questions and revision videos for every GCSE level 9-1 topic! No fees, no trial period, just totally free access to the UKs best GCSE maths revision platform. 100% Free, Forever.View Product
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