Referring to the AP Inter 2nd Year Maths Study Material Chapter 7 Integrals Exercise 7g Solutions makes it easier to understand complex problems.
AP Inter 2nd Year Maths Integrals Solutions Exercise 7g
I.
Question 1.
Find the integral of \(\sqrt{4-x^2}\)
Solution:
Let I = \(\int \sqrt{4-x^2} d x=\int \sqrt{(2)^2-(x)^2} d x\) [∵ \(\int \sqrt{a^2-x^2} d x=\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+C\)]
∴ I = \(\frac{x}{2} \sqrt{4-x^2}+\frac{4}{2} \sin ^{-1} \frac{x}{2}+C=\frac{x}{2} \sqrt{4-x^2}+2 \sin ^{-1} \frac{x}{2}+C\)
Question 2.
Find the integral of \(\sqrt{1-4 x^2}\)
Solution:
Let I = \(\int \sqrt{1-4 \mathrm{x}^2} \mathrm{dx}=\int \sqrt{1^2-(2 \mathrm{x})^2} \mathrm{dx}\) dx Put 2x = t ⇒ 2 dx = dt
∴ I = \(\frac{1}{2} \int \sqrt{1^2-t^2}=\frac{1}{2}\left[\frac{t}{2} \sqrt{1-t^2}+\frac{1}{2} \sin ^{-1} t\right]+C=\frac{t}{4} \sqrt{1-t^2}+\frac{1}{4} \sin ^{-1} t+C\)
= \(\frac{2 x}{4} \sqrt{1-4 x^2}+\frac{1}{4} \sin ^{-1} 2 x+C=\frac{x}{2} \sqrt{1-4 x^2}+\frac{1}{4} \sin ^{-1} 2 x+C\)
Question 3.
Find the integral of \(\sqrt{x^2+4 x+6}\)
Solution:
Question 4.
Find the integral of \(\sqrt{x^2+4 x+1}\)
Solution:
Let I = \(\int \sqrt{x^2+4 x+1} d x=\int \sqrt{\left(x^2+4 x+4\right)-3} d x\)
= \(\int \sqrt{(x+2)^2-(\sqrt{3})^2} d x\) [∵ \(\int \sqrt{x^2-a^2} d x=\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \log \left|x+\sqrt{x^2-a^2}\right|+C\)]
∴ I = \(\frac{(x+2)}{2} \sqrt{x^2+4 x+1}-\frac{3}{2} \log \left|(x-2)+\sqrt{x^2+4 x+1}\right|+C\)
Question 5.
Find the integral of \(\sqrt{1-4 x-x^2}\)
Solution:
Let I = \(\int \sqrt{1-4 x-x^2} d x=\int \sqrt{1-\left(x^2+4 x+4-4\right)} d x=\int \sqrt{1+4-(x+2)^2} d x\)
= \(\int \sqrt{(\sqrt{5})^2-(x+2)^2} d x\) [∵ \(\int \sqrt{a^2-x^2} d x=\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+C\)]
∴ I = \(\frac{(x+2)}{2} \sqrt{1-4 x-x^2}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+C\)
Question 6.
Find the integral of \(\sqrt{x^2+4 x-5}\)
Solution:
Let I = \(\int \sqrt{x^2+4 x-5} d x\)
= \(\int \sqrt{\left(x^2+4 x+4\right)-9} d x=\int \sqrt{(x+2)^2-3^2} d x\)
∴ I = \(\frac{(x+2)}{2} \sqrt{x^2+4 x-5}-\frac{9}{2} \log \left|(x+2)+\sqrt{x^2+4 x-5}\right|+C\)
Question 7.
Find the integral of \(\sqrt{1+3 x-x^2}\)
Solution:
Question 8.
Find the integral of \(\sqrt{x^2+3 x}\)
Solution:
Question 9.
Find the integral of \(\sqrt{1+\frac{x^2}{9}}\)
Solution:
