Question

How do you integrate \[\dfrac{{\tan x}}{{{{(\cos x)}^2}}}\]?

Answer 1

Hint: Here in this question, we have to integrate the given function. The function is in the form of trigonometry. First, we simplify the trigonometric function and then we are applying the integration to the function. Hence, we obtain the required solution for the question.

Complete step-by-step solution:
The function is related to the trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. In trigonometry the cosecant trigonometry ratio is the reciprocal to the sine trigonometry ratio. the secant trigonometry ratio is the reciprocal to the cosine trigonometry ratio and the cotangent trigonometry ratio is the reciprocal to the tangent trigonometry ratio.
The tangent trigonometry ratio is defined as \[\tan x = \dfrac{{\sin x}}{{\cos x}}\] , The cosecant trigonometry ratio is defined as \[\csc x = \dfrac{1}{{\sin x}}\], The secant trigonometry ratio is defined as \[\sec x = \dfrac{1}{{\cos x}}\] and The tangent trigonometry ratio is defined as \[\cot x = \dfrac{{\cos x}}{{\sin x}}\]
Now consider the given function \[\dfrac{{\tan x}}{{{{(\cos x)}^2}}}\], the secant trigonometry ratio is defined as \[\sec x = \dfrac{1}{{\cos x}}\]. Now the function is written as
\[ \Rightarrow \tan x{(\sec x)^2}\]
The squaring of a trigonometry ratio is written as
\[ \Rightarrow \tan x{\sec ^2}x\]
Now we apply the integration to the function
\[ \Rightarrow \int {\tan x{{\sec }^2}x} \,dx\]
Let we substitute \[u = \tan x\], then \[ \Rightarrow du = {\sec ^2}x\,dx\]. Therefore the function is written as
\[ \Rightarrow \int {u\,du} \]
On integrating
\[ \Rightarrow \dfrac{{{u^2}}}{2} + C\], where C is the integration constant.
Substituting the value of u,
\[ \Rightarrow \dfrac{{{{\tan }^2}x}}{2} + C\]
Hence we have integrated the given trigonometric function and obtained the solution.

Thus the final answer of the solution is \[\dfrac{{{{\tan }^2}x}}{2} + C\]

Note: While integrating the trigonometric functions, we simplify the trigonometric functions as much as possible by using the trigonometry ratios or by trigonometry identities. The integration by substitution is the easiest way to integrate. The function and its derivative must be present while substituting.