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Name: Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 39 Maths Textbook Solution.
Uploaded: Sat, 2021-08-21 18:17
Description: Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 39 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 39 Maths Textbook Solution.

Answers (1)

Answer:

$\frac{1}{\sqrt{17}}log\left ( \frac{21+5\sqrt{7}}{4} \right )$

Hint: Use indefinite formula and then put limits to solve this integral

Given:

$\int_{0}^{2}\frac{1}{4+x-x^{2}}dx$

Solution:

$$$ \begin{aligned} &\int_{0}^{2} \frac{1}{4+x-x^{2}} d x=\int_{0}^{2} \frac{1}{-\left(x^{2}-x-4\right)} d x \\ &=-\int_{0}^{2} \frac{1}{x^{2}-2 x \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}-4} d x \\ &=-\int_{0}^{2} \frac{1}{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{1}{4}\right)-4} d x \\ &=-\int_{0}^{2} \frac{1}{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{1+16}{4}\right)} d x \end{aligned}$

$$$ \begin{aligned} &=-\int_{0}^{2} \frac{1}{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{17}{4}\right)} d x \\ &=\int_{0}^{2} \frac{1}{\left(\frac{17}{4}\right)-\left(x-\frac{1}{2}\right)^{2}} d x \end{aligned}$

Putting $\left ( x-\frac{1}{2} \right )=t$

$\Rightarrow dx=dt$

When $x=0$ then

$t=-\frac{1}{2}$

and when $x=2$ then

$t=2-\frac{1}{2}=\frac{4-1}{2}=\frac{3}{2}$

Then

$$$ \begin{aligned} &\int_{0}^{2} \frac{1}{4+x-x^{2}} d x=\int_{-\frac{1}{2}}^{\frac{3}{2}} \frac{1}{\left(\frac{\sqrt{17}}{2}\right)^{2}-t^{2}} d x \\ &=\left[\frac{1}{2 \times \frac{\sqrt{17}}{2}} \log \left|\frac{\frac{\sqrt{17}}{2}+t}{\frac{\sqrt{17}}{2}-t}\right|\right]_{\frac{-1}{2}}^{\frac{3}{2}} \end{aligned}$

$$$ \begin{aligned} &\left.=\frac{1}{\sqrt{17}}\left[\log \left|\frac{\frac{\sqrt{17}}{2}+\frac{3}{2}}{\frac{\sqrt{17}}{2}-\frac{3}{2}}\right|-\log \mid \frac{\frac{\sqrt{17}}{2}+\left(-\frac{1}{2}\right)}{\frac{\sqrt{17}}{2}-\left(-\frac{1}{2}\right)}\right)\right] \\ &=\frac{1}{\sqrt{17}}\left[\log \left|\frac{\frac{\sqrt{17}}{2}+\frac{3}{2}}{\frac{\sqrt{17}-3}{2}}\right|-\log \left|\frac{\frac{\sqrt{17}}{2}+\left(-\frac{1}{2}\right)}{\frac{\sqrt{17}}{2}+\frac{1}{2}}\right|\right] \end{aligned}$