Exercise 2.12 - Chapter 2 Integral Calculus I 12th Maths Guide Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated On May 15, 2024

By SaraNextGen

Ex $2.12$
Choose the correct answer.
Question 1.
$\int \frac{1}{x^3} d x$ is
(a) $\frac{-3}{x^2}+c$
(b) $\frac{-1}{2 x^2}+c$
(c) $\frac{-1}{3 x^2}+c$
(d) $\frac{-2}{x^2}+c$
Answer:
(b) $\frac{-1}{2 x^2}+c$
Hint:
$
\int \frac{1}{x^3} d x=\int x^{-3} d x=\frac{x^{-2}}{-2}+c=\frac{-1}{2 x^2}+c
$
Question 2.
$\int 2^x d x$ is
(a) $2^x \log 2+c$
(b) $2^x+c$
(c) $\frac{2^x}{\log 2}+c$
(d) $\frac{\log 2}{2^x}+c$
Answer:
(c) $\frac{2^x}{\log 2}+c$
Question 3.
$\int \frac{\sin 2 x}{2 \sin x} d x$ is
(a) $\sin x+c$
(b) $\frac{1}{2} \sin \mathrm{x}+\mathrm{c}$
(c) $\cos \mathrm{x}+\mathrm{c}$
(d) $\frac{1}{2} \cos x+c$
Answer:
(a) $\sin \mathrm{x}+\mathrm{c}$
Hint:
$
\int \frac{\sin 2 x}{2 \sin x} d x=\int \frac{2 \sin x \cos x}{2 \sin x} d x=\int \cos x d x=\sin x+c
$

Question 4.
$\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x$ is
(a) $-\cos 2 x+c$
(b) $-\frac{1}{2} \cos 2 \mathrm{x}+\mathrm{c}$
(c) $-\frac{1}{4} \cos 2 x+c$
(d) $-4 \cos 2 x+c$
Answer:
(a) $-\cos 2 x+c$
Hint:
$
\begin{aligned}
\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x & =\int \frac{2 \sin 2 x \cos 3 x}{\cos 3 x} d x=\int 2 \sin 2 x d x \\
& =\frac{-2 \cos 2 x}{2}+c=-\cos 2 x+c
\end{aligned}
$
Question 5.
$\int \frac{\log x}{x} d x, x>0$ is
(a) $\frac{1}{2}(\log x)^2+c$
(b) $-\frac{1}{2}(\log x)^2$
(c) $\frac{2}{x^2}+c$
(d) $\frac{2}{x^2}+c$
Answer:
(a) $\frac{1}{2}(\log x)^2$
Hint:
$
\int \frac{\log x}{x} d x=\int \log x d(\log x)=\frac{(\log x)^2}{2}+c
$
Question 6.
$\int \frac{e^x}{\sqrt{1+e^x}} d x$ is
(a) $\frac{e^x}{\sqrt{1+e^x}}+c$
(b) $2 \sqrt{1+e^x}+c$
(c) $\sqrt{1+e^x}+c$
(d) $e^x \sqrt{1+e^x}+c$
Answer:
(b) $2 \sqrt{1+e^x}+c$

Hint:
$
\int \frac{e^x}{\sqrt{1+e^x}} d x=\int \frac{d\left(1+e^x\right)}{\sqrt{1+e^x}}=2 \sqrt{1+e^x}+c
$
Question 7.
$\int \sqrt{e^x} d x$ is
(a) $\sqrt{e^x}+c$
(b) $2 \sqrt{e^x}+c$
(c) $\frac{1}{2} \sqrt{e^x}+c$
(d) $\frac{1}{2 \sqrt{e^x}}+c$
Answer:
(b) $2 \sqrt{e^x}+c$
Hint:
$
\int \sqrt{e^x} d x=\int \frac{e^x}{\sqrt{e^x}} d x=\int \frac{d\left(e^x\right)}{\sqrt{e^x}}=2 \sqrt{e^x}+c
$
Question 8.
$\int e^{2 x}\left[2 x^2+2 x\right] d x$ is
(a) $e^{2 x} x^2+c$
(b) $x e^{2 x}+c$
(c) $2 x^2 e^2+c$
(d) $\frac{x^2 e^x}{2}+c$
Answer:
(a) $e^{2 x} x^2+c$
Hint:
$\int e^{a x}\left[a f(x)+f^{\prime}(x)\right] d x=e^{a x} f(x)+c$
So $\int e^{2 x}\left[2 x^2+2 x\right] d x=e^{2 x} x^2+c$
Here $a=2, f(x)=x^2, f^{\prime}(x)=2 x$
Question 9.
$\int \frac{e^x}{e^x+1} d x$ is
(a) $\log \left|\frac{e^x}{e^x+1}\right|+c$
(b) $\log \left|\frac{e^x+1}{e^x}\right|+c$
(c) $\log \left|e^x\right|+c$
(d) $\log \left|e^x+1\right|+c$

Answer:
(d) $\log \left|e^x+1\right|+c$
Hint:
$
\int \frac{e^x}{e^x+1} d x=\int \frac{d\left(e^x+1\right)}{e^x+1}=\log \left|e^x+1\right|+c
$
Question 10.
$\int\left[\frac{9}{x-3}-\frac{1}{x+1}\right] d x$ is
(a) $\log |\mathrm{x}-3|-\log |\mathrm{x}+1|+\mathrm{c}$
(b) $\log |x-3|+\log |x+1|+c$
(c) $9 \log |x-3|-\log |x+1|+c$
(d) $91 \log |\mathrm{x}-3|+\log |\mathrm{x}+1|+\mathrm{c}$
Answer:
(c) $9 \log |x-3|-\log |x+1|+c$
Question 11.
$\int \frac{2 x^3}{4+x^4} d x$ is
(a) $\log \left|4+x^4\right|+\mathrm{c}$
(b) $\frac{1}{2} \log \left|4+x^4\right|+\mathrm{c}$
(c) $\frac{1}{4} \log \left|4+x^4\right|+c$
(d) $\log \left|\frac{2 x^3}{4+x^4}\right|+c$
Answer:
(b) $\frac{1}{2} \log \left|4+x^4\right|+c$
Hint:
$
\int \frac{2 x^3}{4+x^4} d x=\frac{1}{2} \int \frac{4 x^3}{4+x^4} d x=\frac{1}{2} \int \frac{d\left(4+x^4\right)}{4+x^4}=\frac{1}{2} \log \left|4+x^4\right|+c
$

Question 12.
$\int \frac{d x}{\sqrt{x^2-36}}$ is............
(a) $\sqrt{x^2-36}+c$
(b) $\log \left|x+\sqrt{x^2-36}\right|+c$
(c) $\log \left|x-\sqrt{x^2-36}\right|+c$
(d) $\log \left|x+\sqrt{x^2-36}\right|+c$
Answer:
(b) $\log \left|x+\sqrt{x^2-36}\right|+c$
Hint:
$
\int \frac{d x}{\sqrt{x^2-36}}=\int \frac{d x}{\sqrt{x^2-6^2}}=\log \left|x+\sqrt{x^2-36}\right|+c
$

Hint:
$
\int \frac{d x}{\sqrt{x^2-36}}=\int \frac{d x}{\sqrt{x^2-6^2}}=\log \left|x+\sqrt{x^2-36}\right|+c
$
Question 13.
$\int \frac{2 x+3}{\sqrt{x^2+3 x+2}} d x$ is
(a) $\sqrt{x^2+3 x+2}+c$
(b) $2 \sqrt{x^2+3 x+2}+c$
(c) $\log \left(x^2+3 x+2\right)+c$
(d) $\frac{2}{3}\left(x^2+3 x+2\right)^{\frac{3}{2}}+c$
Answer:
(b) $2 \sqrt{x^2+3 x+2}+c$
Hint:
$
\int \frac{2 x+3}{\sqrt{x^2+3 x+2}} d x=\int \frac{d\left(x^2+3 x+2\right)}{\sqrt{x^2+3 x+2}}=2 \sqrt{x^2+3 x+2}+c
$
Question 14.
$\int_0^1(2 x+1) d x$ is
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(b) 2
Hint:
$
\int_0^1(2 x+1) d x=\left[x^2+x\right]_0^1=2
$
Question 15.
$\int_2^4 \frac{d x}{x}$ is
(a) $\log 4$
(b) 0
(c) $\log 2$
(d) $\log 8$
Answer:
(c) $\log 2$

Hint:
$
\int_2^4 \frac{d x}{x}=[\log x]_2^4=\log 4-\log 2=\log 2
$
Question 16.
$\int_0^{\infty} e^{-2 x} d x$ is
(a) 0
(b) 1
(c) 2
(d) $\frac{1}{2}$
Answer:
(d) $\frac{1}{2}$
Hint:
$
\int_0^{\infty} e^{-2 x} d x=\left[\frac{e^{-2 x}}{-2}\right]_0^{\infty}=\frac{e^{-\infty}}{-2}-\frac{e^o}{-2}=\frac{0}{-2}+\frac{1}{2}=\frac{1}{2}
$
Question 17.
$\int_{-1}^1 x^3 e^{x^4} d x$ is
(a) 1
(b) $2 \int_0^1 x^3 e^{x^4} d x$
(c) 0
(d) 2
Answer:
(c) 0
Hint:
$
\begin{aligned}
\int_{-1}^1 x^3 e^{x^4} d x & =\frac{1}{4} \int_{-1}^1 e^{x^4}\left(4 x^3\right) d x \\
& =\frac{1}{4} \int_{-1}^1 e^{x^4} d\left(x^4\right)=\left[e^{x^4}\right]_1^{-1}=e-e=0
\end{aligned}
$
(or) Let $f(x)=x^3 e^{x^4}$
$
f(-x)=-x^3 e^{x^4}=-f(x)
$
So $f(x)$ is an odd function. Hence $\int_{-1}^1 f(x) d x=0$

Question 18.
If $\mathrm{f}(\mathrm{x})$ is a continuous function and $\mathrm{a}<\mathrm{c}<\mathrm{b}$, then $\int_a^c f(x) d x+\int_c^b f(x) d x$ is
(a) $\int_a^b f(x) d x-\int_a^c f(x) d x$
(b) $\int_a^c f(x) d x-\int_a^a f(x) d x$
(c) $\int_a^b f(x) d x$
(d) 0
Answer:
(c) $\int_a^b f(x) d x$
Question 19.
The value of $\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x$ is
(a) 0
(b) 2
(c) 1
(d) 4
Answer:
(b) 2
Hint:
$\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x=2 \int_0^{\frac{\pi}{2}} \cos x d x \quad$ (Since $\cos x$ is even) $=2[\sin x]_0^{\frac{\pi}{2}}=2$
Question 20.
$\int_0^1 \sqrt{x^4(1-x)^2} d x$ is
(a) $\frac{1}{12}$
(b) $\frac{-7}{12}$
(c) $\frac{7}{12}$
(d) $\frac{-1}{12}$
Answer:
(a) $\frac{1}{12}$

Hint:
$
\begin{aligned}
\int_0^1 \sqrt{x^4(1-x)^2} d x & =\int_0^1 x^2(1-x) d x \\
& =\int_0^1\left(x^2-x^3\right) d x=\left[\frac{x^3}{3}-\frac{x^4}{4}\right]_0^1=\frac{1}{3}-\frac{1}{4}=\frac{1}{12}
\end{aligned}
$
Question 21.
If $\int_0^1 f(x) d x=1, \int_0^1 x f(x) d x=a$ and $\int_0^1 x^2 f(x) d x=a^2$, then $\int_0^1(a-x)^2 f(x) d x$
(a) $4 a^2$
(b) 0
(c) $2 a^2$
(d) 1
Answer:
(b) 0
Hint:
$
\begin{aligned}
\int_0^1(a-x)^2 f(x) d x & =\int_0^1\left(a^2+x^2-2 a x\right) f(x) d x \\
& =\int_0^1 a^2 f(x) d x+\int_0^1 x^2 f(x) d x-2 a \int_0^1 x f(x) d x \\
& =a^2+a^2-2 a(a)=2 a^2-2 a^2=0
\end{aligned}
$

Question 22.
The value of $\int_2^3 f(5-x) d x-\int_2^3 f(x) d x$ is
(a) 1
(b) 0
(c) $-1$
(d) 5
Answer:
(b) 0
Hint:
$
\begin{aligned}
\int_2^3 f(5-x) d x-\int_2^3 f(x) d x & =\int_2^3 f[5-(3+2-x)] d x-\int_2^3 f(x) d x \\
& =\int_2^3 f(x) d x-\int_2^3 f(x) d x=0
\end{aligned}
$
Question 23.
$\int_0^4\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$ is
(a) $\frac{20}{3}$
(b) $\frac{21}{3}$
(c) $\frac{28}{3}$
(d) $\frac{1}{3}$
Answer:
(c) $\frac{28}{3}$

Hint:
$
\begin{aligned}
& \int_0^4\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x=\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right]_0^4 \\
& \frac{2}{3}\left(4^{\frac{3}{2}}\right)+2\left(4^{\frac{1}{2}}\right)=\frac{2}{3}(8)+4=\frac{28}{3}
\end{aligned}
$
Question 24.
$\int_0^{\frac{\pi}{3}} \tan x d x$ is
(a) $\log 2$
(b) 0
(c) $\log \sqrt{ } 2$
(d) $2 \log 2$
Answer:
(a) $\log 2$
Hint:
$
\begin{aligned}
\int_0^{\frac{\pi}{3}} \tan x d x & =\int_0^{\frac{\pi}{3}} \frac{\sin x}{\cos x} d x=-\int_0^{\frac{\pi}{3}} \frac{-\sin x}{\cos x} d x \\
& =-\int_0^{\frac{\pi}{3}} \frac{d(\cos x)}{\cos x}=[-\log \cos x]^{\frac{\pi}{3}} \\
& =-\log \cos \frac{\pi}{3}+\log \cos 0 \\
& =-\log \left(\frac{1}{2}\right)=-[\log 1-\log 2]=\log 2
\end{aligned}
$
Question 25.
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function $\Gamma(\mathrm{n})$ when $\mathrm{n}=8$ is
(a) 5040
(b) 5400
(c) 4500
(d) 5540
Answer:
(a) 5040

Hint:
$
\Gamma(8)=\Gamma(7+1)=7 !=5040
$
Question 26.
$\Gamma(\mathrm{n})$ is
(a) $(n-1) !$
(b) $\mathrm{n}$ !
(c) $\mathrm{n} \Gamma$ (n)
(d) $(n-1) \Gamma(n)$
Answer:
(a) $(n-1) !$
Hint:
$
\Gamma(\mathrm{n})=\Gamma(\mathrm{n}-1)+1=(\mathrm{n}-1) !
$
Question 27.
$\Gamma(1)$ is
(a) 0
(b) 1
(c) $\mathrm{n}$
(d) $n$ !
Answer:
(b) 1
Hint:
$
\Gamma(1)=(1-1) !=0 !=1
$
Question 28.
If $n>0$, then $\Gamma(n)$ is
(a) $\int_0^1 e^{-x} x^{n-1} d x$
(b) $\int_0^1 e^{-x} x^n d x$
(c) $\int_0^{\infty} e^x x^{-n} d x$
(d) $\int_0^{\infty} e^{-x} x^{n-1} d x$
Answer:
(d) $\int_0^{\infty} e^{-x} x^{n-1} d x$

Question 29.
$\Gamma\left(\frac{3}{2}\right)$ is
(a) $\sqrt{\pi}$
(b) $\frac{\sqrt{\pi}}{2}$
(c) $2 \sqrt{\pi}$
(d) $\frac{3}{2}$
Answer:
(b) $\frac{\sqrt{\pi}}{2}$
Hint:
$
\Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \Gamma\left(\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2}
$
Question 30.
$\int_0^{\infty} x^4 e^{-x} d x$ is
(a) 12
(b) 4
(c) 4!
(d) 64
Answer:
(c) 4!
Hint:
$
\int_0^{\infty} x^4 e^{-x} d x=\int_0^{\infty} x^n e^{-a x} d x=\frac{n !}{a^{n+1}}
$
Here $n=4, a=1=\frac{4 !}{1^5}=4$!