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  • Please solve RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 5 maths textbook solution.

Please solve RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 5 maths textbook solution.

Answers (1)

Answer: \\\frac{4}{3}\left(x^{2}+x+1\right)^{\frac{3}{2}}+\frac{3(2 x+1)}{8} \sqrt{x^{2}+x+1}+\frac{27}{8} \log \left|\left(x+\frac{1}{2}\right)+\sqrt{x^{2}+x+1}\right|+C

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

Given:   I=\int(4 x+1) \sqrt{x^{2}-x-2} d x

Solution: Let, 4 x+1=M \frac{d}{d x}\left(x^{2}-x-2\right)+N

\begin{aligned} &\Rightarrow 4 x+1=M(2 x-1)+N \\ &\Rightarrow 4 x+1=2 M x-M+N \end{aligned}

Now comparing the coefficients of x and the constant term, we get

\begin{aligned} &2 M=4 \Rightarrow M=2 \text { and }-M+N=1 \Rightarrow-2+N=1 \Rightarrow N=3 \\ &I=2 \int(2 x-1) \sqrt{x^{2}-x-2} d x+3 \int \sqrt{x^{2}-x-2} d x \end{aligned}

For first integral, let, x^{2}-x-2=t \Rightarrow(2 x-1) d x=d t

For second integral, x^{2}-x-2=x^{2}-2 \cdot x \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}-2=\left(x-\frac{1}{2}\right)^{2}-\frac{9}{4}=\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}

So, the integral becomes

I=2 \int \sqrt{t} d t+3 \int \sqrt{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}} d x

Use the formula : \left[\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|\right]+C

\begin{aligned} &I=2 \frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+3\left[\frac{\left(x-\frac{1}{2}\right)}{2} \sqrt{x^{2}-x-2}-\frac{\frac{9}{4}}{2} \log \left|\left(x-\frac{1}{2}\right)+\sqrt{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}\right|\right]+C \\ &I=2 \times 2 \frac{t^{\frac{3}{2}}}{3}+\frac{3}{2}\left[\left(x-\frac{1}{2}\right) \sqrt{x^{2}-x-2}-\frac{9}{4} \log \left|\left(x-\frac{1}{2}\right)+\sqrt{x^{2}-x-2}\right|\right]+C \end{aligned}

\\I=\frac{4}{3}\left(x^{2}-x-2\right)^{3 / 2}+\frac{3}{4}(2 x-1)\sqrt{x^{2} x-2}-\frac{27}{8}\log \left|\left(x-\frac{1}{2}\right)+\sqrt{x^{2}-x-2}\right|+C

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