PLEASE!!!!!!!!!!!! how do you find the definite integral from 4 to 5 of x(x^2 + 3y)^(1/2) dy ? ???

Question

zepdrix · Answer

$$\large \large \int\limits_4^5 x \sqrt{x^2+3y}dy$$Hmm, maybe just apply a U-substitution so you can figure out what's going on here :)
Remember that we're treating X as a constant since we're integrating with respect to y.
It will even be constant when we make a substitution.

$$\large \color{orangered}{u=x^2+3y}$$

zepdrix · Answer

$$\large du=0+3 dy$$Understand how that du came about? :D

zepdrix · Answer

Dividing both sides by 3 gives us,
$$\large \color{orangered}{\frac{1}{3}du=dy}$$

If we plug the orange pieces into our integral we get,$$\large \int\limits_{y=4}^5 x \sqrt u \left(\frac{1}{3}du\right)$$

Pulling the constants outside gives us,$$\large \frac{1}{3}x \int\limits_{y=4}^5 u^{1/2}du$$

anonymous · Answer

omg how did you know to set u to that instead of x?? i started to use integration by parts way and its not working T_T

zepdrix · Answer

D:

See how the differential at the end is a $dy$?
So we don't really care much about X in this integral at all since there is no $dx$ present.

So we treat all X's as constants.

zepdrix · Answer

If it helps, think of the integral like this,$$\large \int\limits_4^5 a \sqrt{a^2+3y}\;dy$$Where $a$ is just some constant that we don't have to worry about while integrating. :o

anonymous · Answer

ohh so we can just take x out of the integral before we even start the problem? so the integrand would only be radical x^2 + 3y? is there no use for integration nby parts now lol sorry T.T

zepdrix · Answer

Yah we don't need to do any by-parts c:
That sounds messy anyway :U hah!

anonymous · Answer

ok thank you so much!!!!!! !!!!!