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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.28 Question 3

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.28 Question 3

Answers (1)

Answer:-

\frac{1}{2}(2 x-1) \sqrt{x-x^{2}}+\frac{1}{8} \sin ^{-1}(2 x-1)+c

Hint:-

Add and subtract \frac{1}{4}

Given:-

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

Solution:-

\begin{aligned} &\int \sqrt{x-x^{2}} d x \\\\ &=\int \sqrt{\frac{1}{4}-\frac{1}{4}+x-x^{2}} d x \end{aligned}

 

\begin{aligned} &=\int \sqrt{\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}-x\right)^{2}} d x \\\\ &=\left(\frac{1-2 x}{4}\right) \sqrt{\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}-x\right)^{2}}-\frac{\left(\frac{1}{2}\right)^{2}}{2} \sin ^{-1} \frac{\left(\frac{1-2 x}{4}\right)}{\frac{1}{2}}+c \end{aligned}

Using the formula

\begin{aligned} &\int \sqrt{a^{2}-x^{2}}=\frac{1}{2} x \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+c \\\\ &=\left(\frac{2 x-1}{4}\right) \sqrt{x-x^{2}}+\frac{1}{8} \sin ^{-1}(2 x-1)+c \end{aligned}

 

 

 

 

 

 

Posted by

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