Question
Question: What is the integral of \({\tan ^5}\left( x \right)dx\)?...
What is the integral of tan5(x)dx?
Solution
To find the integral of tan5(x)dx, first of all we have to write tan5(x)dx as tan3(x)×tan2(x). Then we have to use the identity 1+tan2x=sec2x and substitute the value of tan2x in the obtained equation. After that simplify the equation and spate all the integration terms.
Now, we will have to use the property
⇒∫[f(x)]n⋅f′(x)dx=n+1[f(x)]n+1 to find the integral.
Complete step by step solution:
In this question, we have to find the integral of tan5(x)dx.
⇒I=∫tan5(x)dx - - - - - - - - - (1)
Now, we can write tan5(x) as tan3(x)×tan2(x). Therefore, equation (1) becomes,
⇒I=∫tan3x⋅tan2xdx- - - - - - - (2)
Now, we know that
⇒1+tan2x=sec2x ⇒tan2x=sec2x−1
Putting the value of tan2x in equation (2), we get
⇒I=∫tan3x⋅(sec2x−1)dx
⇒I=∫tan3x⋅sec2x−tan3xdx- - - - - - - (3)
Now, again we can write tan3x as tan2x×tanx. Therefore, equation (3) becomes
⇒I=∫tan3x⋅sec2x−tan2x⋅tanxdx- - - - - (4)
Now, again substituting the value of tan2x in equation (4), we get
⇒I=∫((tan3x⋅sec2x)−(sec2x−1)⋅tanx)dx
⇒I=(∫(tan3x⋅sec2x)−(sec2x⋅tanx)+tanx)dx- - - - - - - - (5)
Now, integrating each term separately in equation (5), we get
⇒I=∫(tan3x⋅sec2x)dx−∫(sec2x⋅tanx)dx+∫tanxdx- - - - - (6)
Now, there is a proved theorem, that
⇒∫[f(x)]n⋅f′(x)dx=n+1[f(x)]n+1
Here, in our equation (6) in first integration term,
⇒f(x)=tanx ⇒n=3 ⇒f′(x)=sec2x
Hence,
⇒∫(tan3x⋅sec2x)dx=3+1tan3+1x=4tan4x
And in second integration term,
⇒f(x)=tanx ⇒n=1 ⇒f′(x)=sec2x
Hence,
⇒∫(sec2x⋅tanx)dx=1+1tan1+1x=2tan2x
And, ∫tanxdx=lnsecx
Therefore, equation (6) becomes
⇒I=4tan4x−2tan2x+lnsecx+c
Where, c is the integration constant.
Hence, the integral of tan5(x)dx is 4tan4x−2tan2x+lnsecx+c.
Note:
We cannot directly integrate trigonometric functions with power greater than 1 as there is no direct formula for it. We have to use the relations and formulas to simplify the expression so that we can integrate it easily. So, while finding the integral of trigonometric functions with power greater than 1, always look for the relations that can be used to simplify the expression.