How to Evaluate Trig Integrals and Simplify with Trig Identities?

  • Context: High School 
  • Thread starter Thread starter dx/dy=?
  • Start date Start date
  • Tags Tags
    Integrals Trig
Click For Summary

Discussion Overview

The discussion revolves around evaluating trigonometric integrals, specifically the integral of sin(x)/cos²(x) and related expressions. Participants explore various methods of evaluation, including substitution and the use of trigonometric identities.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant seeks help with the integral ∫(sin x)/(cos² x) dx, expressing difficulty with trigonometric functions.
  • Another participant suggests a substitution method, y = cos x, leading to the integral being transformed into ∫(-1/y²) dy, which evaluates to 1/cos x + C.
  • There is a question about the integral ∫(sec θ)/(cos θ) dx equating to tan θ + C, with references to the definitions of secant and the derivative of tangent.
  • Some participants discuss the possibility of evaluating the integral without substitution, questioning if there exists a function with cos² x as its derivative.
  • One participant mentions using the identity cos(2x) = 2cos² x - 1 to express cos² x in terms of cos(2x), although they express skepticism about its utility in the context of the integral.
  • Another participant states that ∫(sin x)/(cos² x) dx can also be expressed as ∫sec x tan x dx, which evaluates to sec x + C.

Areas of Agreement / Disagreement

Participants generally agree on the substitution method as a valid approach to evaluate the integral, but there is no consensus on the necessity or utility of finding an anti-derivative for cos² x. The discussion remains unresolved regarding the best method to evaluate the integral without substitution.

Contextual Notes

Some participants express uncertainty about the existence of a function whose derivative is cos² x, and there are unresolved questions about the implications of using trigonometric identities in the evaluation process.

Who May Find This Useful

Students and individuals learning calculus, particularly those struggling with trigonometric integrals and identities.

dx/dy=?
Messages
49
Reaction score
0
Can someone please help me in evaluating the following integral?

Im really bad with trig functions, and have been trying to work this out for ages.

[tex]\int \frac{sin x}{cos^2 x} dx[/tex]

Is there a way I can evaluate for [tex]\cos^2 x[/tex] ?
Or is there something I am missing altogether?



Also,

Can someone please tell me how:
[tex]\int \frac{sec\theta}{cos\theta} dx =\ tan\theta + C[/tex]
??
 
Last edited:
Physics news on Phys.org
dx/dy=? said:
[tex]\int \frac{sin x}{cos^2 x} dx[/tex]
Substitution: [itex]y = \cos x \Leftrightarrow dy = - \sin xdx[/itex] gives:

[tex]\int {\frac{{\sin x}}<br /> {{\cos ^2 x}}dx} = \int {\frac{{ - 1}}<br /> {{y^2 }}dy = } \frac{1}<br /> {y} + C = \frac{1}<br /> {{\cos x}} + C[/tex]

dx/dy=? said:
Can someone please tell me how:
[tex]\int \frac{sec\theta}{cos\theta} dx =\ tan\theta = C[/tex]
If you know that :
[tex]\sec \theta = \frac{1}{{\cos \theta }}[/tex]

and

[tex]\frac{{d\left( {\tan \theta } \right)}}<br /> {{d\theta }} = \frac{1}<br /> {{\cos ^2 \theta }}[/tex]

then... :smile:
 
Thanks,

For the first integral [tex]\int \frac{sin x}{cos^2 x} dx[/tex] was thinking more along the lines of multiplying out before evaluatng, to try to obtain a more simple integral to evaluate,
but your way is obviously better.

Is there a way this can be evaluated without substitution?
Or is there no function which has [tex]\cos^2 x[/tex] as its derivative?

Thanks again for the help.
 
Last edited:
why would you want cos^2@ as a derivative?
 
dx/dy=? said:
Is there a way this can be evaluated without substitution?
Or is there no function which has [tex]\cos^2 x[/tex] as its derivative?

Thanks again for the help.
The easiest way is certainly using that substitution.

I don't really see why you'd want to know an anti-derivative for [tex]\cos^2 x[/tex], if you do: just compute it, use:
[tex]\cos \left( {2x} \right) = 2\cos ^2 x - 1 \Leftrightarrow \cos ^2 x = \frac{{1 + \cos \left( {2x} \right)}}<br /> {2}[/tex]

I don't see how that would help though, you don't have [tex]\cos^2 x[/tex] in the integral but [tex]\frac{1}<br /> {{\cos ^2 x}}[/tex], whose anti-derivative is of course [tex]\tan x[/tex]
 
[tex]\int \frac{sin x}{cos^2 x} dx = \int \sec x \tan x dx = \sec x + C[/tex]
 
Thanks very much for all your help everyone.

Im slowly getting the hang of Calculus, but I am not at all good with derivatives of trig functions.
Ill get there eventually.
 

Similar threads

MHB How to Evaluate the Integral of an Even Trig Function in a Radical?
  • · Replies 9 ·
Replies
9
Views
2K
High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
6K
Undergrad Integration Using Trigonometric Substitution
  • · Replies 8 ·
Replies
8
Views
3K
MHB How can I evaluate the integral using a trigonometric identity?
  • · Replies 5 ·
Replies
5
Views
2K
MHB -7.3.89 Integral with trig subst
  • · Replies 4 ·
Replies
4
Views
2K
High School Trigonometric substitution, a case I'd like to share
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Integration of √(1-x^2) with x=sinθ: Check Correctness and Simplification
  • · Replies 12 ·
Replies
12
Views
4K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
4K
MHB 7.3.5 Integral with trig substitution
  • · Replies 1 ·
Replies
1
Views
2K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
4K