How do you integrate #int 2/sqrt(3x-7)# using substitution?

Question

Luke Alvarez · Accepted Answer

#(1/3)(sqrt(3x-7))#

Set #u=3x-7#

That means that #du=3dx#. However, you don't have a 3 on the top.

Don't worry. Instead, just multiply the integral by #3/3#. That way, you have the 3 sufficient for the substitution.

Your equation becomes:

#int (3 * 2)/(3 sqrt(3x-7))dx#

You can then substitute u into the square root and du for 3dx using the three on the top. You should then get

#int 2/(3 sqrtu)#

Now personally, I like to take the #2/3# out from the integral and write #sqrtu# as #u^(-1/2)#. Rewriting this, you get

#2/3 int u^(-1/2)#

Ignore the #2/3# for now and focus on the #int u^(-1/2)#. Add 1 to the power and then multiply by #1/2# since that's the number you get from adding 1 to the power. Simplify and you should have:

#(1/3)(u^(1/2))#.

Now since #u=3x-7#, we need to plug that back into the result. Also, the power of #1/2# is the same thing as the square root. Plugging in and writing a square root, we should get:

#(1/3)(sqrt(3x-7))#

Axel Alvarez · Answer

undefined To integrate $ \int \frac{2}{\sqrt{3x - 7}} $ using substitution, we can let $ u = 3x - 7 $. Then, find $ du $ by differentiating both sides with respect to $ x $, which gives $ du = 3dx $.

Now substitute $ u $ and $ du $ into the integral:

$$ \int \frac{2}{\sqrt{3x - 7}} \, dx $$

Let $ u = 3x - 7 $ and $ du = 3dx $, so $ \frac{du}{3} = dx $.

Substitute $ u $ and $ \frac{du}{3} $ into the integral:

$$ \int \frac{2}{\sqrt{u}} \cdot \frac{du}{3} $$

This simplifies to:

$$ \frac{2}{3} \int \frac{1}{\sqrt{u}} \, du $$

Now integrate $ \frac{1}{\sqrt{u}} $ with respect to $ u $:

$$ \frac{2}{3} \int u^{-1/2} \, du $$

Using the power rule for integration, we get:

$$ \frac{2}{3} \cdot \frac{u^{1/2}}{1/2} + C $$

Simplifying further:

$$ \frac{4}{3} \sqrt{u} + C $$

Finally, substitute back $ u = 3x - 7 $ to get the result in terms of $ x $:

$$ \frac{4}{3} \sqrt{3x - 7} + C $$

So, the integral $ \int \frac{2}{\sqrt{3x - 7}} \, dx $ evaluates to $ \frac{4}{3} \sqrt{3x - 7} + C $, where $ C $ is the constant of integration.