Solve the integral by substitution. \int \frac{dy}{(1+y)\sqrt{y}}

arenceabigns

arenceabigns

Answered question

2021-11-10

Solve the integral by substitution.
dy(1+y)y

Answer & Explanation

broliY

broliY

Skilled2021-11-11Added 97 answers

Step 1
Given: I=dy(1+y)y
For evaluating given integral, we substitute
y=t...(1)
Now, differentiating equation(1) with respect to y
ddy(y)=dtdy   (ddx(x)=12x)
12y=dtdy
dyy=2dt
Step 2
Now, replacing (dyy) with 2dt, y with t in given integral and integrate it
So,
dy(1+y)y=2dt(1+t2)
=2dt1+t2   (dx1+x2=tan1(x)+c)
=2tan1(t)+c   (t=y)
=2tan1(y)+c
Hence, given integral is equal to 2tan1(y)+c.

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