Integral of 1/(1-(x+1)^(1/3)) dx
The solution
Detail solution
-
There are multiple ways to do this integral.
Method #1
-
Let u=3x+1.
Then let du=3(x+1)32dx and substitute −3du:
∫(−u−13u2)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫u−1u2du=−3∫u−1u2du
-
Rewrite the integrand:
u−1u2=u+1+u−11
-
Integrate term-by-term:
-
The integral of un is n+1un+1 when n=−1:
∫udu=2u2
-
The integral of a constant is the constant times the variable of integration:
∫1du=u
-
Let u=u−1.
Then let du=du and substitute du:
∫u1du
-
The integral of u1 is log(u).
Now substitute u back in:
log(u−1)
The result is: 2u2+u+log(u−1)
So, the result is: −23u2−3u−3log(u−1)
Now substitute u back in:
−23(x+1)32−33x+1−3log(3x+1−1)
Method #2
-
Rewrite the integrand:
1−3x+11=−3x+1−11
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−3x+1−11)dx=−∫3x+1−11dx
-
Let u=3x+1.
Then let du=3(x+1)32dx and substitute 3du:
∫u−13u2du
-
The integral of a constant times a function is the constant times the integral of the function:
∫u−1u2du=3∫u−1u2du
-
Rewrite the integrand:
u−1u2=u+1+u−11
-
Integrate term-by-term:
-
The integral of un is n+1un+1 when n=−1:
∫udu=2u2
-
The integral of a constant is the constant times the variable of integration:
∫1du=u
-
Let u=u−1.
Then let du=du and substitute du:
∫u1du
-
The integral of u1 is log(u).
Now substitute u back in:
log(u−1)
The result is: 2u2+u+log(u−1)
So, the result is: 23u2+3u+3log(u−1)
Now substitute u back in:
23(x+1)32+33x+1+3log(3x+1−1)
So, the result is: −23(x+1)32−33x+1−3log(3x+1−1)
-
Now simplify:
−23(x+1)32−33x+1−3log(3x+1−1)
-
Add the constant of integration:
−23(x+1)32−33x+1−3log(3x+1−1)+constant
The answer is:
−23(x+1)32−33x+1−3log(3x+1−1)+constant
The answer (Indefinite)
[src]
/
| 2/3
| 1 3 _______ / 3 _______\ 3*(x + 1)
| ------------- dx = C - 3*\/ x + 1 - 3*log\-1 + \/ x + 1 / - ------------
| 3 _______ 2
| 1 - \/ x + 1
|
/
∫1−3x+11dx=C−23(x+1)32−33x+1−3log(3x+1−1)
The graph
Use the examples entering the upper and lower limits of integration.