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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.4
Integrate the functions in Exercises 86 to 108.
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Question : 88 of 271
Marks: +1, -0
1(2−x)2+1\frac{1}{\sqrt{(2-x)^2+1}}
Solution:  
Let I=dx(2−x)2+1I = \frac{dx}{\sqrt{(2-x)^2+1}}
Put (2−x)=t(2 - x) = t ⇒−dx=dt\Rightarrow -dx = dt ⇒dx=−dt\Rightarrow dx = -dt
∴I=−∫dtt2+1\therefore I = -\int \frac{dt}{\sqrt{t^2+1}} =−log⁡∣t2+1+C∣= -\log \left| \sqrt{t^2+1} + C \right|
=−log⁡∣(2−x)+(2−x)2+1∣+C= -\log \left| (2-x) + \sqrt{(2-x)^2+1} \right| + C
=log⁡∣1(2−x)+x2−4x+5∣+C= \log \left| \frac{1}{(2-x)+\sqrt{x^2-4x+5}} \right| + C
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