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Integration of trig. functions II

paulmdrdo

Active member
May 13, 2013
386
How about these?

∫tan^3xdx

∫(sin^4x cos^2x)dx
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: integration of trig func.

1.) \(\displaystyle \int\tan^3(x)\,dx\)

Rewrite the integrand as \(\displaystyle \tan(x)\tan^2(x)\), then use the Pythagorean identity \(\displaystyle \tan^2(\theta)=\sec^2(\theta)-1\).

2.) \(\displaystyle \int\sin^4(x)\cos^2(x)\,dx\)

I would write the integrand as:

\(\displaystyle \sin^2(x)(\sin(x)\cos(x))^2\)

Now, try applying the double-angle identity for sine on the second factor and the power reduction identity for sine on the first. There will be further work after that, but this should get you started...
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
How about these?

∫tan^3xdx

∫(sin^4x cos^2x)dx
I usually prefer converting to sines and cosines first...

\(\displaystyle \displaystyle \begin{align*} \int{\tan^3{(x)}\,dx} &= \int{\frac{\sin^3{(x)}}{\cos^3{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)}\sin^2{(x)}}{\cos^3{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)} \left[ 1 - \cos^2{(x)} \right] }{\cos^3{(x)}} \, dx} \\ &= \int{ \frac{-\sin{(x)}\left[ \cos^2{(x)} - 1 \right] }{\cos^3{(x)}}\,dx} \end{align*}\)

Now let \(\displaystyle \displaystyle \begin{align*} u = \cos{(x)} \implies du = -\sin{(x)}\,dx \end{align*}\) and the integral becomes

\(\displaystyle \displaystyle \begin{align*} \int{\frac{ -\sin{(x)}\left[ \cos^2{(x)} - 1 \right] }{\cos^3{(x)}}\,dx} &= \int{ \frac{u^2 - 1}{u^3}\,du} \\ &= \int{ \frac{1}{u} - u^{-3}\,du} \\ &= \ln{ |u|} + \frac{1}{2}u^{-2} + C \\ &= \ln{ \left| \cos{(x)} \right| } + \frac{1}{2\cos^2{(x)}} + C \end{align*}\)


As for the second, following Mark's initial suggestion, I would try to use more double angle identities to avoid the power reduction formulae (which I can never remember)...

\(\displaystyle \displaystyle \begin{align*} \int{\sin^4{(x)}\cos^2{(x)}\,dx} &= \int{\sin^2{(x)}\sin^2{(x)}\cos^2{(x)}\,dx} \\ &= \int{ \sin^2{(x)} \left[ \sin{(x)}\cos{(x)} \right] ^2 \, dx} \\ &= \int{\sin^2{(x)} \left[ \frac{1}{2}\sin{(2x)} \right] ^2 \, dx} \\ &= \frac{1}{4} \int{ \sin^2{(x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{4} \int{ \frac{1}{2}\left[ 1 - \cos{(2x)} \right] \sin^2{(2x)}\, dx} \\ &= \frac{1}{8} \int{ \sin^2{(2x)} - \cos{(2x)}\sin^2{(2x)} \, dx} \\ &= \frac{1}{8} \int{ \frac{1}{2} \left[ 1 - \cos{(4x)} \right] - \cos{(2x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{16} \int{ 1\, dx} - \frac{1}{16} \int{ \cos{(4x)}\,dx} - \frac{1}{16} \int{ 2\cos{(2x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{16}x - \frac{1}{16} \left[ \frac{\sin{(4x)}}{4} \right] - \frac{1}{16} \left[ \frac{\sin^3{(2x)}}{3} \right] + C \\ &= \frac{x}{16} - \frac{\sin{(4x)}}{64} - \frac{\sin^3{(2x)}}{48} + C \end{align*}\)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
...I would try to use more double angle identities to avoid the power reduction formulae (which I can never remember)...
We are actually doing the same thing. The power reduction identity I had in mind is:

\(\displaystyle \sin^2(\theta)=\frac{1-\cos(2\theta)}{2}\)
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
We are actually doing the same thing. The power reduction identity I had in mind is:

\(\displaystyle \sin^2(\theta)=\frac{1-\cos(2\theta)}{2}\)
I thought you were referring to the power reduction formula to integrate \(\displaystyle \displaystyle \begin{align*} \sin^{n}{(x)} \end{align*}\), not the double angle identity for cosine...
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775

paulmdrdo

Active member
May 13, 2013
386
I usually prefer converting to sines and cosines first...

\(\displaystyle \displaystyle \begin{align*} \int{\tan^3{(x)}\,dx} &= \int{\frac{\sin^3{(x)}}{\cos^3{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)}\sin^2{(x)}}{\cos^3{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)} \left[ 1 - \cos^2{(x)} \right] }{\cos^3{(x)}} \, dx} \\ &= \int{ \frac{-\sin{(x)}\left[ \cos^2{(x)} - 1 \right] }{\cos^3{(x)}}\,dx} \end{align*}\)

Now let \(\displaystyle \displaystyle \begin{align*} u = \cos{(x)} \implies du = -\sin{(x)}\,dx \end{align*}\) and the integral becomes

\(\displaystyle \displaystyle \begin{align*} \int{\frac{ -\sin{(x)}\left[ \cos^2{(x)} - 1 \right] }{\cos^3{(x)}}\,dx} &= \int{ \frac{u^2 - 1}{u^3}\,du} \\ &= \int{ \frac{1}{u} - u^{-3}\,du} \\ &= \ln{ |u|} + \frac{1}{2}u^{-2} + C \\ &= \ln{ \left| \cos{(x)} \right| } + \frac{1}{2\cos^2{(x)}} + C \end{align*}\)


As for the second, following Mark's initial suggestion, I would try to use more double angle identities to avoid the power reduction formulae (which I can never remember)...

\(\displaystyle \displaystyle \begin{align*} \int{\sin^4{(x)}\cos^2{(x)}\,dx} &= \int{\sin^2{(x)}\sin^2{(x)}\cos^2{(x)}\,dx} \\ &= \int{ \sin^2{(x)} \left[ \sin{(x)}\cos{(x)} \right] ^2 \, dx} \\ &= \int{\sin^2{(x)} \left[ \frac{1}{2}\sin{(2x)} \right] ^2 \, dx} \\ &= \frac{1}{4} \int{ \sin^2{(x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{4} \int{ \frac{1}{2}\left[ 1 - \cos{(2x)} \right] \sin^2{(2x)}\, dx} \\ &= \frac{1}{8} \int{ \sin^2{(2x)} - \cos{(2x)}\sin^2{(2x)} \, dx} \\ &= \frac{1}{8} \int{ \frac{1}{2} \left[ 1 - \cos{(4x)} \right] - \cos{(2x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{16} \int{ 1\, dx} - \frac{1}{16} \int{ \cos{(4x)}\,dx} - \frac{1}{16} \int{ 2\cos{(2x)} \sin^2{(2x)} \, dx} \\ &= \frac{1}{16}x - \frac{1}{16} \left[ \frac{\sin{(4x)}}{4} \right] - \frac{1}{16} \left[ \frac{\sin^3{(2x)}}{3} \right] + C \\ &= \frac{x}{16} - \frac{\sin{(4x)}}{64} - \frac{\sin^3{(2x)}}{48} + C \end{align*}\)

where did you get the sin4x/4 part?
 

paulmdrdo

Active member
May 13, 2013
386
i understood the first 8 lines of your solution but after that i'm stucked! this is what i do..

∫ sin²x (sin²x cos²x) dx

= ∫ sin²x (sinx cosx)² dx

half-angle identity:

sin²x = (1/2)[1 - cos(2x)]

the double-angle identity:

sin(2x) = 2sinx cosx → sinx cosx = (1/2) sin(2x)

the integral becomes:

∫ (1/2)[1 - cos(2x)] [(1/2) sin(2x)]² dx =

∫ (1/2)[1 - cos(2x)] (1/4) sin²(2x) dx =

pulling out the constants and expanding the integrand,

(1/8) ∫ [sin²(2x) - cos(2x) sin²(2x)] dx =

break it into:

(1/8) ∫ sin²(2x) dx - (1/8) ∫ sin²(2x) cos(2x) dx =

(1/8) ∫ (1/2){1 - cos[2(2x)]} dx - (1/8) ∫ sin²(2x) cos(2x) dx =

(1/16) ∫ dx - (1/16) ∫ cos(4x) dx - (1/8) ∫ sin²(2x) cos(2x) dx = ---> this is where i'm stuck how can the cos4x be sin4x/4? please help me!
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Let's look at:

\(\displaystyle \int\cos(4x)\,dx\)

Now, if we let:

\(\displaystyle u=4x\,\therefore\,du=4\,dx\)

then we have:

\(\displaystyle \frac{1}{4}\int\cos(u)\,du\)

Now, integrate, then back-substitute for $u$, and you have the result.
 

Petrus

Well-known member
Feb 21, 2013
739
i understood the first 8 lines of your solution but after that i'm stucked! this is what i do..

∫ sin²x (sin²x cos²x) dx

= ∫ sin²x (sinx cosx)² dx

half-angle identity:

sin²x = (1/2)[1 - cos(2x)]

the double-angle identity:

sin(2x) = 2sinx cosx → sinx cosx = (1/2) sin(2x)

the integral becomes:

∫ (1/2)[1 - cos(2x)] [(1/2) sin(2x)]² dx =

∫ (1/2)[1 - cos(2x)] (1/4) sin²(2x) dx =

pulling out the constants and expanding the integrand,

(1/8) ∫ [sin²(2x) - cos(2x) sin²(2x)] dx =

break it into:

(1/8) ∫ sin²(2x) dx - (1/8) ∫ sin²(2x) cos(2x) dx =

(1/8) ∫ (1/2){1 - cos[2(2x)]} dx - (1/8) ∫ sin²(2x) cos(2x) dx =

(1/16) ∫ dx - (1/16) ∫ cos(4x) dx - (1/8) ∫ sin²(2x) cos(2x) dx = ---> this is where i'm stuck how can the cos4x be sin4x/4? please help me!
Hello Paul,
indeed that \(\displaystyle \cos(4x)\neq \frac{\sin(4x)}{4}\) BUT what prove it did Was that he integrate it! Let's check if it is correct! So if we derivate \(\displaystyle \frac{\sin(4x)}{4}\) we shall get \(\displaystyle \cos(4x)\) so if we use chain rule to derivate that we get \(\displaystyle \frac{4\cos(4x)}{4}\) and if we simplifie that we get \(\displaystyle \cos(4x)\) which we wanted to get:)
Edit:Mark Was faster:(
Regards,
\(\displaystyle |\pi\rangle\)
 

paulmdrdo

Active member
May 13, 2013
386
Let's look at:

\(\displaystyle \int\cos(4x)\,dx\)

Now, if we let:

\(\displaystyle u=4x\,\therefore\,du=4\,dx\)

then we have:

\(\displaystyle \frac{1}{4}\int\cos(u)\,du\)

Now, integrate, then back-substitute for $u$, and you have the result.
no i understand that part.. my follow up question is how would i deal with the third term in the integrand ((1/8) ∫ sin²(2x) cos(2x) dx )?

should i use the power reduction formula for sin²(2x) again?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
no i understand that part.. my follow up question is how would i deal with the third term in the integrand ((1/8) ∫ sin²(2x) cos(2x) dx )?

should i use the power reduction formula for sin²(2x) again?
You asked (twice):

where did you get the sin4x/4 part?
...how can the cos4x be sin4x/4? please help me!
So this is what I was addressing. To evaluate:

\(\displaystyle \int\sin^2(2x)\cos(2x)\,dx\)

I would use the substitution:

\(\displaystyle u=\sin(2x)\,\therefore\,du=2\cos(2x)\,dx\)

and so we have:

\(\displaystyle \frac{1}{2}\int u^2\,du\)

Now, integrate, then back-substitute for $u$. :D
 

paulmdrdo

Active member
May 13, 2013
386
You asked (twice):





So this is what I was addressing. To evaluate:

\(\displaystyle \int\sin^2(2x)\cos(2x)\,dx\)

I would use the substitution:

\(\displaystyle u=\sin(2x)\,\therefore\,du=2\cos(2x)\,dx\)

and so we have:

\(\displaystyle \frac{1}{2}\int u^2\,du\)

Now, integrate, then back-substitute for $u$. :D
hmmm..correct me if my thougth processes is wrong.

(1/8) ∫ [sin²(2x) - cos(2x) sin²(2x)] dx = in this part of the solution can i use that substitution of u=sin2x? in that way i don't have to use the power reduction formula anymore. please bear with me.
 

Petrus

Well-known member
Feb 21, 2013
739
hmmm..correct me if my thougth processes is wrong.

(1/8) ∫ [sin²(2x) - cos(2x) sin²(2x)] dx = in this part of the solution can i use that substitution of u=sin2x? in that way i don't have to use the power reduction formula anymore. please bear with me.
Hello Paul,
Just a fast respond as I have to think more when I am home but that Dont work, Cause you Will have \(\displaystyle du\) on one side... You need on both side:)
Regards,
\(\displaystyle |\pi\rangle\)
 

Petrus

Well-known member
Feb 21, 2013
739
hmmm..correct me if my thougth processes is wrong.

(1/8) ∫ [sin²(2x) - cos(2x) sin²(2x)] dx = in this part of the solution can i use that substitution of u=sin2x? in that way i don't have to use the power reduction formula anymore. please bear with me.
Hello Paul,
I think now this should work
subsitute \(\displaystyle u=2x <=> du=2\) and integrate them separate so you got
\(\displaystyle \frac{1}{16}\int\sin^2(u) du-\frac{1}{16}\int\cos(u)\sin^2(u)du\)
does this make it simpler?

Regards
\(\displaystyle |\pi\rangle\)