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A lvl H2 Maths: Integration

Question from http://www.sgforums.com/forums/2297/topics/339990

(a) Evaluate


(b) Using the substitution , find

with x in the range 0 < x < 1.

(c) Find \int \frac{2x-1}{x^{2}-2x+5}dx.

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Answer:

(a)




Note: For , the numerator is a differential of the denominator.


(b)
From ,


Sub and






Since ,
Also, \cos^{2} u = 1 - \sin^{2} u
\cos u = \sqrt{1 - \sin^{2} u}
\cos u = \sqrt{1 - x}

Thus, sin 2u = 2 sin u cos u
\sin 2u = 2\sqrt{x}\sqrt{1-x}
\sin 2u = 2\sqrt{x-x^{2}}

Hence,
=\sin^{-1} \sqrt{x}-\sqrt{x-x^{2}}+c


(c)
\int \frac{2x-1}{x^{2}-2x+5}dx
\small =\int \frac{2x-2}{x^{2}-2x+5}+\frac{1}{x^{2}-2x+5}dx
\small =\int \frac{2x-2}{x^{2}-2x+5}dx+\int \frac{1}{x^{2}-2x+5}dx
\small =\int \frac{2x-2}{x^{2}-2x+5}dx+\int \frac{1}{\left(x-1\right)^{2}+4}dx
\small =\int \frac{2x-2}{x^{2}-2x+5}dx+\frac{1}{4}\int \frac{1}{\left(\frac{x-1}{2}\right)^{2}+1}dx
\small =\ln \left( x^{2}-2x+5\right)+\frac{1}{4}\tan^{-1}\left(\frac{x-1}{2}\right)\bullet 2+c
\small =\ln \left( x^{2}-2x+5\right)+\frac{1}{2}\tan^{-1}\left(\frac{x-1}{2}\right)+c



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