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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.28 Question 1

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.28 Question 1

Answers (1)

Answer:-

\frac{1}{2}(x-1) \sqrt{3+2 x-x^{2}}+2 \sin ^{-1}\left(\frac{x-1}{2}\right)+c

Hint:-

Adding and subtracting with 1.

Given:-

\int \sqrt{3+2 x-x^{2}} d x

Solution:-

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

\begin{aligned} &=\frac{1}{2}(x-1) \sqrt{3+2 x-x^{2}}+\frac{1}{2} \times(2)^{2} \sin ^{-1}\left(\frac{x-1}{2}\right)+c \\\\ &=\frac{1}{2}(x-1) \sqrt{3+2 x-x^{2}}+2 \sin ^{-1}\left(\frac{x-1}{2}\right)+c \end{aligned}

 

Using the formula

 

\int \sqrt{a^{2}+x^{2}} d x=\frac{1}{2} x \sqrt{a^{2}+x^{2}}+\frac{1}{2} a^{2} \sin ^{-1}\left(\frac{x}{2}\right)+c

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