find the following indefinite integral?

let u = 1 - 3x^2 ---> answer = -(2/3)√(1 - 3x^2 ) + c

2x / sqrt(1 - 3x^2) dx

I'm going to factor the 2 outside of the integral. Thus our integrand (the thing to the right of the integral sign), is just:

x / sqrt(1 - 3x^2) dx

Say that:

u = 1 - 3x^2

Calculate the derivative of u:

du = -6x * dx

Now if we want to substitute u into our integrand, we must make sure we also have a corresponding value of du. We currently only have x*dx. So if we multiply the integrand by -6, we'll have -6x*dx, which is equal to du. BUT, we cannot just manipulate one half, we must equally and oppositely do the same on the other side. Thus we multiply by -1/6 on the outside. If we were to combine this, it would be the same as multiplying the equation by -6/-6, or just +1 (which, as we know, does not change the value at all):

-2/6 * ∫ du/sqrt(u)

At this point, you can, if you want, substitute back in du = -6xdx and u=1-3x^2 to see that I did not fundamentally change the problem at all.

Now we can calculate 1/sqrt(u)'s antiderivative easily with a power rule, knowing that this is just u^(-1/2). Thus we get:

2u^(1/2) + c

where c is an arbitrary constant of integration.

Then remember to multiply out the scalar -2/6 we had in front and resubstitute u = 1 - 3x^2:

(-1/3)(2(1 - 3x^2)^(1/2)) + c

It isn't important we multiply c by the -1/3 as we can pick any number for c and get a valid answer.

Simplifying:

(-2/3) * sqrt(1 - 3x^2) + c

take out the 2 in front for and u subsitution inside the radical...u=1-3x^2 ==> -1/6du=xdx so its now

-1/3 ∫1/ √u and thats -1/3 *2 √u +c back subsitute for -2/3√(1-3x^2) +c

A) ?(2x-2)/ (2x^2-4x + a million)^3) dx do u-substitution u = 2x^2-4x + a million du = 4x -4 dx (a million/2) ? a million/ u^3 du (a million/2) ? u^-3 du (a million/2) (-a million/2) (u^-2) (-a million/4)(u^-2) = (-a million/4)(a million/u^2) + c replace lower back u = 2x^2-4x + a million (-a million/4)(a million/(2x^2-4x + a million)^2) + c rewrite: answer: (-a million/(4(2x^2-4x + a million)^2)) + c B) ?(e^x-7x^2)/(4) dx (a million/4)? (e^x-7x^2)dx (a million/4)? e^x dx - (7/4)? x^2 dx critical of ? e^x dx = e^x critical of ? x^2 dx = x^3 / 3 (a million/4)e^x - (7/4)(a million/3)(x^3) + c answer: (a million/4)(e^x) - (7/12)x^3 + c desire this facilitates!