Solve the integral by substitution. \int (z^{2}+2z+1)^{\frac{4}{3}}dz

CheemnCatelvew

CheemnCatelvew

Answered question

2021-11-05

Solve the integral by substitution.
(z2+2z+1)43dz

Answer & Explanation

Fatema Sutton

Fatema Sutton

Skilled2021-11-06Added 88 answers

Step 1
The given integral is,
I=(z2+2z+1)43dz
On simplifying the term under integral, we get
z2+2z+1=(z)2+2z1+(1)2
=(z+1)2.[a2+2ab+b2=(a+b)2]
Hence, the integral becomes,
I=((z+1)2)43dz
=(z+1)83dz
Step 2
On substituting z+1 as t
z+1=t
dz=dt
Hence, the given integral becomes,
I=t83dt
Using the power rule of integration, we get
I=t83dt
=[t83+183+1]+C
=[t113(113)]+C
=3t11311+C
Step 3
Putting back the value of t, we get
3t11311+C=3(z+1)11311+C
Therefore, the expression for the given integral is 3(z+1)11311+C

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