Question

Evaluate the following integral:
$\int {{{\sec }^2}x\,dx} $
$\left( A \right)\,\,2\tan x + C$
$\left( B \right)\,\,\tan 2x + C$
$\left( C \right)\,\,\tan x + C$
$\left( D \right)\,\,None\,of\,these$

Answer 1

Hint: This question is just an application of a single formula of integration. We can also get the answer if we know when we get ${\sec ^2}x$ as our answer while doing differentiation of some function. One can also try by converting the function in $\sin x\,or\,\cos x$ and then on further calculation get the required answer.

Complete step-by-step answer:
In the given question, we know that
On differentiating $\tan x$ with respect to x, we get ${\sec ^2}x$
$\frac{{d\left( {\tan x} \right)}}{{dx}} = {\sec ^2}x$
On cross-multiplication, we get
$d\left( {\tan x} \right) = {\sec ^2}x\,dx$
Now, integrating both sides
$\tan x + C\, = \,\int {{{\sec }^2}xdx} $
 Here C is the constant of integration.
Therefore, our required answer is $\tan x + C$.
So, the correct answer is “Option C”.

Note: An integral which is not having any upper and lower limit is known as an indefinite integral. Mathematically, if F(x) is any anti-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted, $\int {f\left( x \right)\,dx = F\left( x \right) + C} $ .Anti derivatives or integrals of the functions are not unique. There exist infinitely many antiderivatives of each of certain functions, which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason, C is customarily referred to as an arbitrary constant. C is the parameter by which one gets different antiderivatives (or integrals) of the given function.