Definite integral by change of variable

In this video, I showed how to use u substitution to evaluate a definite integral

[farkliyahya]

I'm starting to binge watch your videos

[KaivalyaChess]

sir you will always be the best math yt channel

[IrvinMoonga-rwul]

Whoever made you a lecturer is a genius!❤

[mondayizuchukwu]

U re gifted in teaching maths how i wish u handle physics as well

[Amoeby]

For a general case of the integral of (x^m)(a + bx^n)^p where m, n and p are rational numbers and a and b are real numbers:
1) if p is an integer then substitute x = t^(LCM of the denominators of m and n);
2) if (m + 1)/n is integer then substitute a + bx^n = t^q where q is the denominator of p;
3) if (m + 1)/n + p is an integer then substitute b + a/x^n = t^q where q is tge denominator of p.

So after rewriting this integral in the binomial expression we get the integral from 0 to 3sqrt(3)/2 of (x^3)(9 + 4x^2)^(3/2)dx.
m = 3, n = 2, p = -3/2.

1) p is not an integer
2) (m + 1)/n = (3 + 1)/2 = 2 is an integer so 9 + 4x^2 = t^2
4xdx = tdt
xdx = tdt/4
x^2 = (t^2 - 9)/4
x^3dx = t(t^2 - 9)dt/16
x = 0 -> t = 3, x = 3sqrt(3)/2 -> t = 6.

Finally, the integral from 0 to 3sqrt(3)/2 of (x^3)(9 + 4x^2)^(3/2)dx is equal to the integral from 3 to 6 of t(t^2 - 9)dt/(16t^3) = (1 - 9t^(-2))dt/16 = (t + 9/t)/16 where t goes from 3 to 6 so it equals to (6 + 9/6 - 3 - 9/3)/16 = 3/32.

[holyshit]

We can factor from numerator derivative of the inside of denominator and we are left with x^2 whch can be expressed as function of sqrt(4x^2+9)
so change of variable u=sqrt(4x^2+9) is good idea

[AliHassan-hbbn]

You could use integration by parts method

[odumosuadeniyilukman]

Sir, I want to ask that condition will I find the value of the limits, or is it every time I have definite integrals, I will take their new value of u.

Coz, I don't understand that.
I just want to know the condition behind it.


🙇

[pgray]

I get the same result without the x^3=x^2 * x trick. At one point I get [sqrt(u-9)]^3 in the numerator and sqrt(u-9) in the denominator. After cancellation left with [sqrt(u-9)]^2 which conveniently is just u-9.

[johnpaul]

Bro. Super. Can contact you personally?

[eliacoldwar-usqr]

Find d²y/dx² when y=square root of x/(x²+x)

[harshplayz]

Its just 1st step game
Aftee u know u have break x³ to x² and x.
After it its easy