find the following indefinite integral .. any help?

Integrals have a nice property called linearity which will help tremendously here. ∫(4 + 9 cos(x) + sin^3(x)dx) = ∫4dx + ∫9 cos(x)dx + ∫sin^3(x)dx. Also, you can factor out a scalar (any old real number without a factor of x or trig function or anything) from an integrand. So the numerator of that expression can be written 4∫dx + 9∫cos(x)dx + ∫sin^3(x)dx.

Anyway, the real pain in the *** about this problem is that ∫sin^3(x)dx. I can go through the derivation if you like, but any table of integrals (or Google) will tell you that ∫sin^3(x)dx is -1/3*sin^2(x)*cos(x) - 2/3*cos(x). +C, of course.

So yeah, that's that, then. Don't forget your constant of integration, and don't forget that after you evaluate each of those integrals, you have to sum them again and stick them back on top of that sin^2(x). Cheers!

2x / sqrt(a million - 3x^2) dx i visit ingredient the two exterior of the crucial. subsequently our integrand (the element to the appropriate of the crucial sign), is only: x / sqrt(a million - 3x^2) dx Say that: u = a million - 3x^2 Calculate the derivative of u: du = -6x * dx Now if we'd pick to replace u into our integrand, we ought to verify we also have a corresponding fee of du. We at the instant purely have x*dx. So if we multiply the integrand through -6, we will have -6x*dx, it relatively is comparable to du. yet, we can not only manage one a million/2, we ought to the two and oppositely do a similar on the different section. subsequently we multiply through -a million/6 on the exterior. If we've been to combine this, it may be comparable to multiplying the equation through -6/-6, or only +a million (which, as all of us understand, does not replace the value in any respect): -2/6 * ? du/sqrt(u) At this factor, you may, in case you pick, replace back in du = -6xdx and u=a million-3x^2 to verify that i did not basically replace the subject in any respect. Now we are in a position to calculate a million/sqrt(u)'s antiderivative certainly with a potential rule, understanding that it relatively is only u^(-a million/2). subsequently we get: 2u^(a million/2) + c the place c is an arbitrary consistent of integration. Then you ought to multiply out the scalar -2/6 we had in front and resubstitute u = a million - 3x^2: (-a million/3)(2(a million - 3x^2)^(a million/2)) + c this is not significant we multiply c through the -a million/3 as we are in a position to %. any huge style for c and get a valid answer. Simplifying: (-2/3) * sqrt(a million - 3x^2) + c

break it up into 3 integrals....show that you attempted the problem at least instead of asking a million questoins...

= -4cotx - 9/sinx - cosx + c