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Problem 68 Hard Difficulty

Evaluate the definite integral.

$ \displaystyle \int^4_0 \frac{x}{\sqrt{1 + 2x}} \, dx $

Answer

$\frac{10}{3}$

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Video Transcript

given our problem, the first thing we know is that acts can be substituted to be you minus one over two or 1/2 times you minus one, which means R D X. Taking the derivative is simply do you divide by two and 1/2 to you? Which means the limits of integration not changed from 042129 Because given the new DX that we have established when you plug in, we're gonna have to do some algebraic manipulation before we can take the integral. First thing we can do is you can pull up the constant of 1/4. We can rewrite it as you to the 1/2 minus you to the negative one house, which means now we can take the integral use. The power rule, which means increased the expert by 123 over to 1/2 plus two over two is three over to do the same thing for this one. What's through? Write this in terms of square roots. It makes a lot easier to plug it using the fundamental theme of calculus. Plug in. This is equivalent to three plus 1/3 which is 10 birds