Question

Which of the following is/are true?
(where $C$ is constant of integration)

• A. $\int a^{3x}dx=\frac{a^{3x}}{3\ln a}+C$
• B. $\int e^{2x}dx=\frac{e^{2x}}{2}+C$
• C. $\int \text{cosec}(x+2)dx=\ln |\text{cose}c(x+2)+\cot (x+2)|+C$
• D. $\int \sec (3x+4)dx=-\ln |\sec (3x+4)+\tan (3x+4)|+C$

Answer 1

Answer: (A), (B)

$\int e^{2x}dx=\frac{e^{2x}}{2}+C$

$\left(a\right)\int a^{3x}dx=\frac{a^{3x}}{\ln a}\times \frac{1}{3}+C$
$(\because \underset{}{\overset{}{\int }}{a}^{bx}dx=\frac{a^{bx}}{\ln a}\times \frac{1}{b}+C;a>0,b\ne 0)$

$\left(b\right)\int e^{2x}dx=\frac{e^{2x}}{2}+C$
$(\because \int e^{ax}dx=\frac{e^{ax}}{a}+C,a\ne 0)$

$\left(c\right)\int \text{cosec}(x+2)dx=-\ln |\text{cosec}(x+2)+\cot (x+2)|+C$
$(\because \int \text{cosec}(ax+b)dx=\frac{-\ln |\text{cosec}(ax+b)+\cot (ax+b)|}{a}+C,a\ne 0)$

$\left(d\right)\int \sec (3x+4)dx=\frac{\ln |\sec (3x+4)+\tan (3x+4)|}{3}+C$
$(\because \int \sec (ax+b)dx=\frac{\ln |\sec (ax+b)+\tan (ax+b)|}{a}+C,a\ne 0)$