help me to integrate this :
integration of (3 sin^2 x cos x) dx
i confuse how to integrate this..

Question

help me to integrate this :
integration of (3 sin^2 x cos x) dx
i confuse how to integrate this..

shubhamsrg · Answer

let sinx = t

kainui · Answer

Use "u"-substitution since you see a function and its derivative next to it.

anonymous · Answer

attachment

kainui · Answer

www.wolframalpha.com show me steps

anonymous · Answer

^^shown

australopithecus · Answer

you have to use identities to solve this

$$\int\limits_{}^{}3 \sin^2 x \cos x$$

kainui · Answer

...

australopithecus · Answer

$$3\int\limits_{}^{}\sin^2 x \cos xdx$$
$$3\int\limits_{}^{}(1-\cos^{2}(x)) \cos xdx$$

kainui · Answer

No, sin^2(x)=u

1/2 du=cosxdx

3/2udu becomes your new integral.

australopithecus · Answer

f(x) = sin^2(x)
f'(x) = cos(x)^2*sin(x)

anonymous · Answer

Why not 
u = sinx --> du = cosxdx ??

anonymous · Answer

= 3 ∫ u² du   = ....

australopithecus · Answer

$$3(\int\limits_{}^{}\cos(x)dx - \int\limits_{}^{}\cos^2(x)dx)$$

anonymous · Answer

It's crystal clear about the relation between sinx and cosxdx :)

australopithecus · Answer

remember for the future

$$\cos^2(x) = \frac{1 + \cos(2x)}{2}$$


so we have,

$$3(\sin(x) + c - \int\limits_{}^{}\frac{1+\cos(2x)}{2}dx)$$

anonymous · Answer

^done in 3rd reply

australopithecus · Answer

You can solve it my way too which is good practice for when you get integrals with trig functions to high powers

shubhamsrg · Answer

@Australopithecus its cos^3 x there and not square

australopithecus · Answer

oh sorry made a mistake lol

anonymous · Answer

where the best solution? i still confuse.. hu2

australopithecus · Answer

it can still be solved with that method

shubhamsrg · Answer

and sinx= t ,u whatever, thats the easiest thing which you can do..as i pointed out in the very beginning ..

shubhamsrg · Answer

cos^3 x will still involve substitution ..

australopithecus · Answer

$$3(\sin(x) + c - \int\limits_{}^{}\cos(x)\frac{1 + \cos(2x)}{2}dx$$

australopithecus · Answer

I'm not arguing that substitution will be involved in my solution but it still works for solving this integral it is just the long way

australopithecus · Answer

I think I made a mistake again though meh

kainui · Answer

@Australopithecus It's like suggesting someone make a tunnel through a mountain when you can just use a teleporter.

australopithecus · Answer

It is still applicable and stop hassling me I'm rusty ha

anonymous · Answer

so, the best solution? anyone?

anonymous · Answer

3rd

anonymous · Answer

@sha0403, to clarify, you should use any method provided except @Australopithecus'

anonymous · Answer

ok thanks u all for helping me..i appreciate it.. =)