Question

How do you integrate 3cos^2*5xdx?

Answer 1

\[\large{\int 3\cos^2 (?) * 5x dx}\]

What is the angle of cos^2 here ?

Answer 2

@vishweshshrimali5 I think he means
\[\int\limits_{}^{} 3( \cos ^{2} (5x))dx\]

Answer 3

Possible :)

Then:

\[\large{I = \int 3\cos^2 (5x) \ dx}\tag{1}\]

First of all substitute:
\[\large{u = 5x \implies du = 5dx}\tag{2}\]

Answer 4

We will get:

\[\large{I = \cfrac{3}{5}\int \cos^2 u \ du}\]

Answer 5

Watching and learning as you're doing it.

Answer 6

Use this:

\[\large{\cos (2\theta) = 2\cos^2 {\theta} - 1}\]

\[\large{\implies 2\cos^2 \theta = \cos(2\theta) + 1}\]

\[\large{\implies \cos^2 \theta = \cfrac{\cos(2\theta) + 1}{2}}\]

Answer 7

oh double angle identity?

Answer 8

Now using this we get:

\[\large{I = \cfrac{3}{5} \int \cfrac{\cos (2u) + 1}{2} \ du}\]

\[\large{\implies I = \cfrac{3}{10} \int [\cos (2u) + 1] \ du}\]

@study100 Yes :)

Answer 9

Ok so far I'm understanding this :)) <3
why abs value?

Answer 10

wait, that sign is not abs. Sorry kinda blind.

Answer 11

Now we can simplify I as:

\[\large{I = \cfrac{3}{10}\left[\int \cos(2u) \ du + \int du \right]}\]

\[\large{\implies I = \cfrac{3}{10}\left[\cfrac{\sin (2u)}{2} + u\right] + C}\]

Answer 12

you're amazing. : o

Answer 13

Now we replace u by 5x:

\[\large{\boxed{\color{green}{I = \cfrac{3}{10}\left[\cfrac{\sin (10x)}{2} + 5x\right] + C}}}\]

Answer 14

@study100 Thanks :)

Answer 15

So using a little bit substitution and trigonometric identities we can solve this question.

Answer 16

I wish I can medal this like 20 times.

Answer 17

:) Thanks

Answer 18

Thanks a lot, this really helped