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  • Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.28 Question 17 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.28 Question 17 Maths Textbook Solution.

Answers (1)

best_answer

Answer:-

\frac{x-1}{2} \sqrt{x^{2}-2 x}-\frac{1}{2} \log \left|x-1+\sqrt{x^{2}-2 x}\right|+c

Hint:-

\begin{aligned} &\int \sqrt{a^{2}-x^{2}}=\frac{1}{2} x \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1}\left(\frac{x}{a}\right)+c \\\\ &\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+c \\\\ &\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+c \end{aligned}

 

Given:-

 

\int \sqrt{x^{2}-2 x} d x

Solution:-

Let,  I=\int \sqrt{x^{2}-2 x} d x

We have

\begin{aligned} &I=\int \sqrt{x^{2}-2 x} d x \\ &I=\int \sqrt{x^{2}-2 x+1^{2}-1^{2}} d x \\ &I=\int \sqrt{(x-1)^{2}-(1)^{2}} d x \end{aligned}

 

As I match with the form

\begin{aligned} &\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+c \\\\ &I=\frac{x-1}{2} \sqrt{(x-1)^{2}-1}-\frac{1}{2} \log \left|x-1+\sqrt{(x-1)^{2}-1}\right|+c \\\\ &I=\frac{x-1}{2} \sqrt{x^{2}-2 x}-\frac{1}{2} \log \left|x-1+\sqrt{x^{2}-2 x}\right|+c \end{aligned}

 

 

 

 

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