Question

can someone show me what is the next step of this integral?

Answer 1

\[\pi \int\limits_{0}^{\frac{ \pi }{ 4 }}(cosx+1)^{2}-(sinx+1)^{2}dx\]

Answer 2

expand the squares
cos^2x - sin^2x = cos 2x
other terms are standard

Answer 3

what do you mean?

Answer 4

(a+b)^2 = a^2+2ab +b^2

(cos x +1)^2 =....

(sin x +1)^2 = ...

Answer 5

oh the top equals sin^2 x and the bottom cos^2 x right?

Answer 6

not actually

(cos x + 1)^2 = cos^2x + 2 cos x +1
no simplification

similarly do (sin x +1)^2

Answer 7

sin^2x+2sinx+1

Answer 8

yes,
so your function to be integrated will be
cos^2x - sin ^2 x + 2 cos x - 2 sin x

right ??

Answer 9

yea

Answer 10

theres still a pi in the front right?

Answer 11

yes, let that pi remain there :)

now use cos^2 x - sin ^2 x = cos 2x

now can you integrate each term ?

cos 2x
2cos x
-2sin x

Answer 12

would it be sin2x+2sinx+2cosx

Answer 13

sin2x /2 isn't it ?

2sin x +2 cos x are correct :)

Answer 14

what is cos2x?

Answer 15

cos 2x = cos^2 x - sin^2 x

integral of cos 2x = (sin 2x)/2

Answer 16

was that your doubt ?

Answer 17

yea

Answer 18

so could you solve the entire integral now ?

Answer 19

am i doing this correctly so far? \[\pi(\frac{-3\sqrt{2} }{ 2}+2)\]

Answer 20

pi [ (1/2 +2/sqrt 2 + 2/sqrt 2) - (0+0+2) ]

how did you get that ?

Answer 21

isn't 1/2sin2(pi/4)=sqrt2/2

Answer 22

1/2 sin (pi/2) = 1/2 (1) = 1/2

because sin pi/2 = 1

Answer 23

oh i thought i plug in pi/4?

Answer 24

you do plug in pi/4

in sin 2x -----> sin 2(pi/4) = sin (2pi/4) = sin (pi/2)

Answer 25

oh i see and how come 2sin(pi/4)=2/sqrt2

Answer 26

sin pi/4 = 1/ sqrt 2

Answer 27

is the answer pi/2?

Answer 28

pi [ (1/2 +2/sqrt 2 + 2/sqrt 2) - (0+0+2) ] = pi[2 sqrt 2 - 3/2]

Answer 29

so that would be the answer?

Answer 30

yes :)

Answer 31

thank you for your help!!!

Answer 32

welcome ^_^