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Solution - Prove that ∫√(x^2-a^2)dx=x/2 √(x^2-a^2)-a^2/2 log|x+√t(x^2-a^2)|+c - Methods of Integration - Integration by Parts

Question

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`

 

Solution

Let  `I=int sqrt(x^2-a^2)dx`

`I=int sqrt(x^2-a^2).1.dx`

`I=sqrt(x^2-a^2).intdx -int[d/dx(sqrt(x^2-a^2))intdx]dx`

`I=xsqrt(x^2-a^2)-int[(2x)/(2sqrt(x^2-a^2))x]dx`

`I=xsqrt(x^2-a^2)-int[(x^2)/(sqrt(x^2-a^2))]dx`

`I=xsqrt(x^2-a^2)-int[(x^2-a^2+a^2)/(sqrt(x^2-a^2))]dx`

`I=xsqrt(x^2-a^2)-int(x^2-a^2)/(sqrt(x^2-a^2))dx+a^2intdx/(sqrt(x^2-a^2)`

`I=xsqrt(x^2-a^2)-intsqrt(x^2-a^2)dx+a^2intdx/(sqrt(x^2-a^2)`

`I=xsqrt(x^2-a^2)-I+a^2intdx/(sqrt(x^2-a^2)`

`2I=xsqrt(x^2-a^2)+a^2log|x+sqrt(x^2-a^2)|+C'`

`I=(xsqrt(x^2-a^2))/2+a^2/2log|x+sqrt(x^2-a^2)|+C^'/2`

`I=(xsqrt(x^2-a^2))/2+a^2/2log|x+sqrt(x^2-a^2)|+C`

 

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APPEARS IN

2014-2015 (March)
Question 5.2.2 | 4 marks
2013-2014 (March)
Question 6.2.2 | 4 marks
Solution for question: Prove that ∫√(x^2-a^2)dx=x/2 √(x^2-a^2)-a^2/2 log|x+√t(x^2-a^2)|+c concept: Methods of Integration - Integration by Parts. For the courses HSC Arts, HSC Science (Computer Science), HSC Science (Electronics), HSC Science (General)
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