Question

int. calculus
\[\int\limits_{}^{}\sin ^{2}x \cos ^{2}x \tan ^{2} x dx\]

Answer 1

Write them as
sin squared x = (1/2)(1 - cos 2x)
cos squared x = (1/2)(1 + cos 2x)

maybe..that'll be easier for you to integrate.

Answer 2

can i do that, even if sin or cos is not raised two?

Answer 3

doesn't that simplify to sin^4 x when tan is written as sin/cos ?

Answer 4

Write tan^2 x = sin^2 x / cos^2 x

Answer 5

yea

Answer 6

i did that before

Answer 7

\[\large{\int { sin^4x dx}}\]

Can you do this?

Answer 8

then i lost any of ideas after that

Answer 9

OK no problem, where are you lost?

Answer 10

can u integrate sin^2 x ?

Answer 11

after i got the \[\int\limits_{?}^{?}\sin ^{4}xdx\]

Answer 12

Well if integration of sin x dx= -cosx + C
then that of sin^4 x dx = ?

Answer 13

-cos^4x + C ??

Answer 14

write sin^4 x = (sin^2 x)(1-cos^2 x)

Answer 15

no, its not that direct

Answer 16

then distribute?

Answer 17

= (sin^2 x)(1-cos^2 x) = sin^2 x - sin^2x cos^2 x
yes

Answer 18

= sin^2 x - sin^2x cos^2 x = sin^2 x - (1/4) (sin 2x)^2

Answer 19

how did you get (1/4)(sin 2x)^2?

Answer 20

sin 2x = 2 sin x cos x
to get sin^2x cos^2 x , i squared the above .....

Answer 21

got that ? ^
and can u integrate sin^2 x - (1/4) (sin 2x)^2

Answer 22

can you show me the whole process of that sin 2x = 2 sin x cos x

Answer 23

u want me to prove 2sin x cos x = sin 2x
or u wanted this :
sin 2x = 2 sin x cos x
(sin 2x)^2 = (2 sin x cos x)^2 = 4 sin^2x cos^2 x
so
sin^2 x cos^2 x = (1/4) (sin 2x)^2

Answer 24

so if u can integrate sin^2 x
you can also integrate sin^2 x - (1/4) (sin 2x)^2