# Integration question, u substitution

• jonc1258
In summary, the problem is evaluating the indefinite integral ∫(√x/(√x-3))dx. The first attempt at a solution involved substituting u for √x-3, but the resulting du did not have a suitable multiple in the integrand. The next attempt involved splitting the integral and using the substitution u=√x-3, which simplifies the problem.
jonc1258
The problem statement
Evaluate the indefinite integral

∫$\frac{\sqrt{x}}{\sqrt{x}-3}$dx

The attempt at a solution
My first thought was to substitute u for √(x)-3, but then du would equal $\frac{1}{2\sqrt{x}}$dx, and there's no multiple of du in the integrand.

Next, I tried splitting up the problem like this:
∫$\frac{1}{\sqrt{x}-3}$*$\sqrt{x}$
When u is substituted for the $\sqrt{x}$, du still equals$\frac{1}{2\sqrt{x}}$dx , but the minus 3 in the denominator of the first factor prevents cleanly substituting a multiple of du for $\frac{1}{\sqrt{x}-3}$dx.

Any ideas or hints?

Notice that if ##u = \sqrt{x} - 3##, then ##u + 3 = \sqrt{x}##.

Since ##\displaystyle du = \frac{1}{2\sqrt{x}} dx, 2\sqrt{x}\ du = dx##

Does this make things easier?

$\sqrt{x}dx= (x/\sqrt{x})dx= (\sqrt(x)^2)(dx/\sqrt{x})= (u+3)^2du$

The denominator is kind of hard to see, it's sqrt(x)-3. Thanks for all the replies, I'll see if I can figure it out

Last edited:
jonc1258 said:
The problem statement
Evaluate the indefinite integral

∫$\frac{\sqrt{x}}{\sqrt{x}-3}$dx

The attempt at a solution
My first thought was to substitute u for √(x)-3, but then du would equal $\frac{1}{2\sqrt{x}}$dx, and there's no multiple of du in the integrand.
If $\displaystyle \ \ du=\frac{1}{2\sqrt{x}}dx=\frac{1}{2u}dx\,,$

then $\displaystyle \ \ dx=2u\,du\ .$

Next, I tried splitting up the problem like this:
∫$\frac{1}{\sqrt{x}-3}$*$\sqrt{x}$
When u is substituted for the $\sqrt{x}$, du still equals$\frac{1}{2\sqrt{x}}$dx , but the minus 3 in the denominator of the first factor prevents cleanly substituting a multiple of du for $\frac{1}{\sqrt{x}-3}$dx.

Any ideas or hints?

## What is integration?

Integration is a mathematical operation that involves finding the area under a curve in a graph. It is essentially the reverse of differentiation, which is finding the slope of a curve at a given point.

## What is u substitution in integration?

U substitution, also known as the method of substitution, is a technique used to simplify and solve integrals. It involves substituting a new variable, u, for a complicated expression within the integral, making it easier to solve.

## When should u substitution be used in integration?

U substitution should be used when an integral involves a complicated expression that cannot be easily solved using other integration techniques, such as integration by parts or trigonometric substitution.

## What are the steps for using u substitution in integration?

The steps for using u substitution in integration are: 1) Identify a complicated expression within the integral, 2) Substitute a new variable, u, for that expression, 3) Rewrite the integral in terms of u, 4) Solve the integral using u as the variable, 5) Substitute back in the original variable to get the final answer.

## What are some common mistakes to avoid when using u substitution in integration?

Some common mistakes to avoid when using u substitution in integration are: 1) Forgetting to substitute back in the original variable at the end, 2) Choosing the wrong substitution or failing to recognize when u substitution should be used, 3) Making mistakes when differentiating or integrating the expression with u, 4) Forgetting to add the constant of integration at the end.

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