Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

How do you integrate 1tan2x with respect to x ?

Answer
VerifiedVerified
467.4k+ views
like imagedislike image
Hint: In this question we have to find the integral of the given function, first multiply and divide with sec2x then assume the tanx as sinθ , and then simplify the expression by using partial fractions, then we will get an expression that is easy to be integrated, so apply integration formula we will get the required value for the given expression.

Complete step by step answer:
Given 1tan2x and we have to integrate with respect to x ,
Let,
 I=1tan2x ,
Multiply and divide with sec2x , we get,
I=1tan2x×sec2xsec2xdx ,
Now we know that, sec2x=1+tan2x now substitute the value in the expression we get,
I=1tan2x1+tan2x(sec2x)dx ,
Let us consider sinθ=tanx ,
Now perform differentiate on both sides, we get,
ddθsinθ=ddxtanx ,
Now deriving we get,
cosθdθ=sec2xdx ,
Now the given expression becomes,
I=1sin2θ1+sin2θcosθdθ ,
We know that 1sin2θ=cos2θ , then the expression becomes,
I=cos2θ1+sin2θcosθdθ ,
Now simplifying we get,
I=cosθ1+sin2θcosθdθ ,
Now multiplying we get,
I=cos2θ1+sin2θdθ ,
Now writing in terms of tanθ and secθ , we get,
I=1sec2θ1+tan2θsec2θdθ ,
Now simplifying we get,
I=1sec2θ+tan2θdθ ,
Now we know that 1+tan2θ=sec2θ , we get,
I=11+tan2θ+tan2θdθ ,
Now simplifying we get,
I=11+2tan2θdθ ,
Now multiply and divide by sec2θ , we get,
I=11+2tan2θ×sec2θsec2θdθ ,
Now using the identity 1+tan2θ=sec2θ , we get,
I=11+2tan2θ×sec2θ1+tan2θdθ ,
Now simplifying we get,
I=sec2θ(1+tan2θ)(1+2tan2θ)dθ ,
Let us consider u=tanθ ,
Now deriving on both sides we get,
du=sec2θdθ ,
Now the expression becomes,
I=1(1+u2)(1+2u2)du ,
Now using performing partial fractions, we get, rewrite the expression we get,
A1+u2+B1+2u2=1(1+u2)(1+2u2) ,
Now taking L.C.M and simplifying we get,
A+2Au2+B+Bu2(1+u2)(1+2u2)=1(1+u2)(1+2u2) ,
Now simplifying we get,
u2(2A+B)+(A+B)(1+u2)(1+2u2)=1(1+u2)(1+2u2) ,
Now comparing the two sides we get,
 2A+B=0(1) and
 A+B=1(2) ,
Now solving the two equations we get,
A=1 and B=2 ,
Now substitute the values in the expression we get,
I=21+2u211+u2du ,
Now separating the terms we get,
I=21+2u2du11+u2du ,
Now we can say that the above terms are in form of tan1 integrals,
I=221+2u2du11+u2du ,
So, from the integral formula we get,
I=2tan1(2u)tan1u ,
Now we know that u=tanθ , substitute the value in the expression we get,
I=2tan1(2tanθ)tan1(tanθ) ,
Now simplifying we get,
I=2tan1(2tanθ)θ ,
Now we know that sinθ=tanx ,
Now apply sin1 on both sides we get,
sin1(sinθ)=sin1(tanx) ,
Now simplifying we get,
θ=sin1(tanx) ,
And we know that tanθ=sinθcosθ , and cosθ=1sin2θ , so we get,
tanθ=sinθ1sin2θ ,
Now we know that sinθ=tanx , by substituting we get,
tanθ=tanx1tan2x ,
Now the expression becomes,
I=2tan1(2tanx1tan2x)sin1(tanx)+C ,
Now substituting the value of I we get,
1tan2xdx=2tan1(2tanx1tan2x)sin1(tanx)+C ,
So, the value of the given expression is,
 2tan1(2tanx1tan2x)sin1(tanx)+C ,

Final Answer:
  The value of the given expression 1tan2xdx is equal to 2tan1(2tanx1tan2x)sin1(tanx)+C .


Note: In this type of question we use both derivation and integration formulas as both are related to each other students should not get confused where to use the formulas and which one to use as there are many formulas in both derivation and integration. Some of the important formulas that are used are given below:
 ddxx=1 ,
 ddxxn=nxn1 ,
 ddxuv=udvdx+vdudx ,
 ddxex=ex ,
 ddxuv=vdudxudvdxv2 ,
 ddxlnx=1x ,
 ddxsinx=cosx ,
 ddxcosx=sinx ,
 ddxtanx=sec2x ,
 dx=x+c ,
 xndx=xn+1n+1+c ,
 1xdx=lnx+c ,
 11+x2dx=tan1x .
Latest Vedantu courses for you
Grade 10 | CBSE | SCHOOL | English
Vedantu 10 CBSE Pro Course - (2025-26)
calendar iconAcademic year 2025-26
language iconENGLISH
book iconUnlimited access till final school exam
tick
School Full course for CBSE students
PhysicsPhysics
Social scienceSocial science
ChemistryChemistry
MathsMaths
BiologyBiology
EnglishEnglish
₹41,000 (9% Off)
₹37,300 per year
Select and buy