Best solutions to - "Solve the integral by substitution. ∫dy(1+y)y"

arenceabigns

Answered question

2021-11-10

Solve the integral by substitution.

\int \frac{d y}{(1 + y) \sqrt{y}}

broliY

Skilled2021-11-11Added 97 answers

Step 1
Given:

I = \int \frac{d y}{(1 + y) \sqrt{y}}

For evaluating given integral, we substitute

\sqrt{y} = t

...(1)
Now, differentiating equation(1) with respect to y

\frac{d}{d y} (\sqrt{y}) = \frac{d t}{d y} (∵ \frac{d}{d x} (\sqrt{x}) = \frac{1}{2 \sqrt{x}})

\frac{1}{2 \sqrt{y}} = \frac{d t}{d y}

\frac{d y}{\sqrt{y}} = 2 d t

Step 2
Now, replacing

(\frac{d y}{\sqrt{y}})

with 2dt,

\sqrt{y}

with t in given integral and integrate it
So,

\int \frac{d y}{(1 + y) \sqrt{y}} = \int \frac{2 d t}{(1 + t^{2})}

= 2 \int \frac{d t}{1 + t^{2}} (∵ \int \frac{d x}{1 + x^{2}} = \tan^{- 1} (x) + c)

= 2 \tan^{- 1} (t) + c (∵ t = \sqrt{y})

= 2 \tan^{- 1} (\sqrt{y}) + c

Hence, given integral is equal to

2 \tan^{- 1} (\sqrt{y}) + c

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