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  • Please Solve RD Sharma Class 12 Chapter Indefinite Integrals Exercise 18.17 Question 1 Maths Textbook Solution

Please Solve RD Sharma Class 12 Chapter Indefinite Integrals Exercise 18.17 Question 1 Maths Textbook Solution

Answers (1)

best_answer

Answer:-\sin^{-1}\left ( x-1 \right )+c

Hint:- To solve this  problem, use special formula.

Given:- \int \frac{1}{\sqrt{2x-x^{2}}}dx

Solution:-

Let

 \begin{aligned} I &=\int \frac{1}{\sqrt{2 x-x^{2}}} d x=\int \frac{1}{\sqrt{-(x^{2}-2 x})} d x \\\\ &\int \frac{1}{\sqrt{-(x-1)^{2}+1}} d x\quad \quad\quad\left [ \because a^{2}-2ab+b^{2}=(a-b)^{2} \right ]\end{aligned}                          

  \begin{aligned} &\text { Put } x-1=t \Rightarrow d x=d t \text { then } \\\\ &\quad \begin{array}{l} I=\int \frac{1}{\sqrt{1^{2}-t^{2}}} d t\quad\left[\because \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x=\sin ^{-1}\left(\frac{x}{a}\right)+c\right] \\ \\ =\sin ^{-1}\left(\frac{t}{1}\right)+c \\\\ =\sin ^{-1}(x-1)+c\quad \quad\quad\left [ \because x-1=t \right ]\\ \end{array} \end{aligned}                                                     

 

                                                                                                       

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