Find the integral by algebraic substitution: \int \frac{dx}{\sqrt{1+4x}}

Tyra

Tyra

Answered question

2021-11-06

Find the integral by algebraic substitution:
dx1+4x

Answer & Explanation

odgovoreh

odgovoreh

Skilled2021-11-07Added 107 answers

Step 1 
In some cases integrals can be simplified to standard integrals by using an appropriate substitution. For example the integral tan(x)sec2(x) dx  can be simplified to udu using the substitution u=tan(x)
This is the case for the given integral. Find an appropriate substitute to simplify the integral. 
Step 2 
The given integral is  dx 1+4x. Use the substitution 1+4x=u. Taking its derivative gives 21+4x dx =du. This can be written as  dx 1+4x=du2. Use this substitution in the integral and integrate. 
 dx 1+4x=du2 
=u2+C 
=1+4x2+C 
Hence, the given integral evaluates to 1+4x2+C.

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