Question

How many of the following are matched correctly?

INTEGRAND	INTEGRAL
$\int x^{n}dx$	$\frac{x^n}{n+1}+C$
$\int \frac{1}{x}dx$	$\ln \left(x\right|+C$
$\int e^{x}dx$	$e^x+C$
$\int a^{x}dx$	$\frac{a^x}{\ln a}+C$
$\int \cos xdx$	$\sin x+C$
$\int \sin xdx$	$\cos x+C$
$\int \sec x\tan xdx$	$\sec x+C$
$\int \mathrm{cosec}x\cot xdx$	$-\mathrm{cosec}x+C$
$\int {\sec }^{2}xdx$	$\tan x+C$
$\int {\mathrm{cosec}}^{2}xdx$	$\cot x+C$

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The correct option is B 7

We, know that integration is the opposite of differentiation.
To verify which of the following integrands are correctly matched, we will differentiate integrals to see whether it matches to the integrand or not.
A.
$\int x^{n}dx=\frac{x^n}{n+1}+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\frac{x^n}{n+1}+C\right)=\frac{n{x}^{n-1}}{n+1}\ne L.H.S\mathrm{Hence},\mathrm{not}\mathrm{true}.$
B.
$\int \frac{1}{x}dx=\ln \left(x\right|+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\ln \left(x\right|+C\right)=\frac{1}{x}=L.H.S\mathrm{Hence},\mathrm{true}.$
C.
$\int e^{x}dx=e^x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(e^x+C\right)=e^x=L.H.S\mathrm{Hence},\mathrm{true}.$
D.
$\int a^{x}dx=\frac{a^x}{\ln a}+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\frac{a^x}{\ln a}+C\right)=\frac{a^x\times \ln a}{\ln a}=a^x=L.H.S\mathrm{Hence},\mathrm{true}.$
E.
$\int \cos xdx=\sin x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\sin x+C\right)=\cos x=L.H.S\mathrm{Hence},\mathrm{true}.$
F.
$\int \sin xdx=\cos x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\cos x+C\right)=-\sin x\ne L.H.S\mathrm{Hence},\mathrm{not}\mathrm{true}.$
G.
$\int \sec x\tan xdx=\sec x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\sec x+C\right)=\sec x\tan x=L.H.S\mathrm{Hence},\mathrm{true}.$
H.
$\int \mathrm{cosec}x\mathrm{cotx}dx=-\mathrm{cosec}x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(-\mathrm{cosec}x+C\right)=\mathrm{cosec}x\mathrm{cotx}=L.H.S\mathrm{Hence},\mathrm{true}.$
I.
$\int {\sec }^{2}xdx=\tan x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\tan x+C\right)={\sec }^{2}x=L.H.S\mathrm{Hence},\mathrm{true}.$
J.
$\int {\mathrm{cosec}}^{2}xdx=\cot x+C\mathrm{Differentiating}R.H.S\mathrm{we}\mathrm{get},\frac{d}{dx}\left(\cot x+C\right)=-{\mathrm{cosec}}^{2}x\ne L.H.S\mathrm{Hence},\mathrm{not}\mathrm{true}.$
Thus, 7 integrals of particular integrands are correct.