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# Integration

## Question

Hello everybody!

May I ask you a big favor? I need help. What I have is: $(2x-3)/\sqrt{(4x-1)}$ What is the integral of it??

I’ve tried to use Integral Calculator here: http://www.numberempire.com/

And I’ve got : $(x-4)*\sqrt{(4*x-1)}/3$

Seems like it is the right answer. But I want to understand the way of solving it. Please show me step by step solution.

Thanks in advance!

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## Comments

## Old comments

Re: Integration

Dear mathstudent200, Please remove your article ”integration” – your question is no encyclopedia entry! The integrand is a linear polynomial divided by the square root of another linear polynomial. You could execute the division by first writing the dividend in the form $(1/2)[(4x-1)-5]$; divide separately the minuend and the subtrahend. Then it’s easy to integrate separately both quotients! Greetings, Jussi

Re: Integration

Usually, if you have an integrand that contains a problematic nested function, like the polynomial inside the square root, we can usually simplify it greatly by letting a single variable u represent the value of that function, and rewriting the whole integral in terms of $u$. In order to integrate the new expression written in terms of $u$, we also need $du/dx$.

In your case, the problematic function is the square root of a polynomial in $x$, so we let $u=4x-1$ in order to turn the problematic square root into a simple power of a single variable $u^{{1/2}}$. We need to write $2x-3$ in terms of $u$ as well. We can solve for $x$ in terms of $u$ by using our equation $u=4x-1$. This gives us $x=(u+1)/4$, so $2x-3$ becomes $(1/2)u-5/2$.

Dividing by $u^{{1/2}}$, our integrand is now $(1/2)u^{{1/2}}-(5/2)u^{{-1/2}}dx$. We cannot integrate this function of $u$ over $dx$, so we must find out how $dx$ is related to $du$. Differentiating $u=4x-1$ gives us $du/dx=4$, or $dx=(1/4)du$. Replacing $dx$ with $(1/4)du$ in our integral allows us to integrate it, as it is just a sum of two power rules. Once we have our integrated function in terms of $u$, we can replace $u$ with $4x-1$ and simplify to get our function in terms of $x$.