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Need solution for RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 14 maths textbook solution.

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Answer : \frac{-2}{9}\left(10-4 x-3 x^{2}\right)^{\frac{3}{2}}+\frac{11}{8}(3 x+2) \sqrt{10-4 x+3 x^{2}}+\frac{187}{9 \sqrt{3}} \sin ^{-1}\left(\frac{3 x+2}{\sqrt{34}}\right)+C

Hint: To solve the given integration,  we express the linear team as a derivative of quadratic into constant plus another constant

Given:  \int(2 x+5) \sqrt{10-4 x-3 x^{2}} d x

Solution : \int(2 x+5) \sqrt{10-4 x-3 x^{2}} dx

\begin{aligned} &\text { Let, }(2 x+5)=A \frac{d}{d x}\left(10-4 x-3 x^{2}\right)+B \\ &(2 x+5)=A(-4-6 x)+B \\ &(2 x+5)=-4 A+B-6 A x \end{aligned}

Comparing the coefficient of x and the constant terms, we get

\begin{aligned} &6 A=-2 \Rightarrow A=-\frac{1}{3} \text { and } \\ &-4 A+B=5 \Rightarrow B=5+4 A=5+4\left(-\frac{1}{3}\right)=\frac{11}{3} \\ &I=-\frac{1}{3} \int(-4-6 x) \sqrt{10-4 x-3 x^{2}} d x \\ &I=-\frac{1}{3} \int(-6 x-4) \sqrt{10-4 x-3 x^{2}} d x \end{aligned}

\begin{aligned} &I=-\frac{1}{3}\left[\int(-6 x-4) \sqrt{10-4 x-3 x^{2}} d x+\int \frac{11}{3} \sqrt{10-4 x-3 x^{2}} d x\right] \\ &I=-\frac{1}{3} \int(-6 x-4) \sqrt{10-4 x-3 x^{2}} d x+\left(\frac{11}{3}\right) \int \sqrt{10-4 x-3 x^{2}} d x \end{aligned}

For the first integral:

\begin{aligned} &\text { Let, } 10-4 x-3 x^{2}=t \\ &\Rightarrow(-6 x-4) d x=d t \\ &I=-\frac{1}{3} \int \sqrt{t} d t+\frac{11}{3} \times \sqrt{3} \int \sqrt{\frac{10}{3}-\frac{4}{3} x-x^{2}} d x \end{aligned}

Use the formula : \int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+C

And \left[\int x^{n} d x=\frac{x^{n}+1}{n+1}+C\right]

\begin{aligned} &I=-\frac{1}{3} \frac{t^{\frac{3}{2}}}{\frac{3}{2}}+\frac{11}{3} \sqrt{3} \int \sqrt{\left(\frac{10}{3}\right)-\left\{x^{2}+2 \frac{2}{3} x+\left(\frac{2}{3}\right)^{2}\right\}+\left(\frac{2}{3}\right)^{2}} d x \\ &I=-\frac{2}{9} t^{3 / 2}+\frac{11}{\sqrt{3}} \int \sqrt{\left(\frac{\sqrt{34}}{3}\right)^{2}-\left(x+\frac{2}{3}\right)^{2}} d x \end{aligned}

\begin{aligned} &I=-\frac{2}{9}\left(10-4 x-3 x^{2}\right)^{\frac{3}{2}}+\frac{11}{\sqrt{3}}\left[\frac{1}{2}\left(x+\frac{2}{3}\right) \sqrt{\left(\frac{\sqrt{34}}{3}\right)^{2}-\left(x+\frac{2}{3}\right)^{2}}+\frac{34 / 9}{2} \sin ^{-1}\left[\frac{x+\frac{2}{3}}{\frac{\sqrt{34}}{3}}\right]\right]+C \\ &I=-\frac{2}{9}\left(10-4 x-3 x^{2}\right)^{\frac{3}{2}}+\frac{11}{\sqrt{3}}\left[\frac{\left(x+\frac{2}{3}\right)\left(\frac{10}{3}-\frac{4 x}{3}-x^{2}\right)^{\frac{1}{2}}}{2}\right]+\frac{187}{9 \sqrt{3}} \sin ^{-1}\left[\frac{3 x+2}{\sqrt{34}}\right]+C \end{aligned}

\\I=-\frac{2}{9}\left(10-4 x-3 x^{2}\right)^{\frac{3}{2}}+\frac{11}{18}(3 x+2) \sqrt{10-4 x-3 x^{2}}+\frac{187}{9 \sqrt{3}} \sin ^{-1}\left[\frac{3 x+2}{\sqrt{34}}\right]+C

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