Integral of tan^3x dx
The solution
Detail solution
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Rewrite the integrand:
tan3(x)=(sec2(x)−1)tan(x)
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There are multiple ways to do this integral.
Method #1
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Let u=sec2(x).
Then let du=2tan(x)sec2(x)dx and substitute 2du:
∫4uu−1du
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The integral of a constant times a function is the constant times the integral of the function:
∫2uu−1du=2∫uu−1du
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Rewrite the integrand:
uu−1=1−u1
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Integrate term-by-term:
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The integral of a constant is the constant times the variable of integration:
∫1du=u
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The integral of a constant times a function is the constant times the integral of the function:
∫(−u1)du=−∫u1du
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The integral of u1 is log(u).
So, the result is: −log(u)
The result is: u−log(u)
So, the result is: 2u−2log(u)
Now substitute u back in:
−2log(sec2(x))+2sec2(x)
Method #2
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Rewrite the integrand:
(sec2(x)−1)tan(x)=tan(x)sec2(x)−tan(x)
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Integrate term-by-term:
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Let u=sec2(x).
Then let du=2tan(x)sec2(x)dx and substitute 2du:
∫41du
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The integral of a constant times a function is the constant times the integral of the function:
∫21du=2∫1du
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The integral of a constant is the constant times the variable of integration:
∫1du=u
So, the result is: 2u
Now substitute u back in:
2sec2(x)
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The integral of a constant times a function is the constant times the integral of the function:
∫(−tan(x))dx=−∫tan(x)dx
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Rewrite the integrand:
tan(x)=cos(x)sin(x)
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Let u=cos(x).
Then let du=−sin(x)dx and substitute −du:
∫u1du
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−u1)du=−∫u1du
-
The integral of u1 is log(u).
So, the result is: −log(u)
Now substitute u back in:
−log(cos(x))
So, the result is: log(cos(x))
The result is: log(cos(x))+2sec2(x)
Method #3
-
Rewrite the integrand:
(sec2(x)−1)tan(x)=tan(x)sec2(x)−tan(x)
-
Integrate term-by-term:
-
Let u=sec2(x).
Then let du=2tan(x)sec2(x)dx and substitute 2du:
∫41du
-
The integral of a constant times a function is the constant times the integral of the function:
∫21du=2∫1du
-
The integral of a constant is the constant times the variable of integration:
∫1du=u
So, the result is: 2u
Now substitute u back in:
2sec2(x)
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−tan(x))dx=−∫tan(x)dx
-
Rewrite the integrand:
tan(x)=cos(x)sin(x)
-
Let u=cos(x).
Then let du=−sin(x)dx and substitute −du:
∫u1du
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−u1)du=−∫u1du
-
The integral of u1 is log(u).
So, the result is: −log(u)
Now substitute u back in:
−log(cos(x))
So, the result is: log(cos(x))
The result is: log(cos(x))+2sec2(x)
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Add the constant of integration:
−2log(sec2(x))+2sec2(x)+constant
The answer is:
−2log(sec2(x))+2sec2(x)+constant
The answer (Indefinite)
[src]
/
| 2 / 2 \
| 3 sec (x) log\sec (x)/
| tan (x) dx = C + ------- - ------------
| 2 2
/
2log(sin2x−1)−2sin2x−21
The graph
1 1
- - + --------- + log(cos(1))
2 2
2*cos (1)
2log(1−sin21)−2sin21−21−21
=
1 1
- - + --------- + log(cos(1))
2 2
2*cos (1)
log(cos(1))−21+2cos2(1)1
Use the examples entering the upper and lower limits of integration.