# Should I memorize all these trigonometric integrals?

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1. May 27, 2015

### Byeonggon Lee

I only memorized these trigonometric differential identities :
sin(x) = cos(x)
cos(x) = -sin(x)

because I can convert tan(x) to sin(x) / cos(x) and
sec(x) to 1 / cos(x) .. etc

And there is no need to memorize some integral identities such as :
∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C

because I memorized
sin(x) = cos(x)
cos(x) = -sin(x)

But these identities seem inevitable to memorize:
∫ sec^2(x) dx = tan(x) + C
∫ cosec^2(x) dx = -cot(x) + C
∫ sec(x)tan(x) dx = sec(x) + C
∫ cosec(x)cot(x) dx = -cosec(x) + C

For example
∫ sec^2(x) dx = tan(x) + C

First I tried to convert sec^2(x) to 1 / cos^2(x)
∫ sec^2(x) dx = ∫ (1 / cos^2(x)) dx

And that's where I'm stuck.
It looks impossible to proceed anymore without memorizing a trigonometric differential identity
`tan(x) = sec^2(x)

Last edited: May 27, 2015
2. May 28, 2015

### Svein

How about remembering how to derive a fraction? $\frac{d}{dx}(\frac{\sin x}{\cos x})=\frac{\sin 'x\cdot \cos x - \cos 'x\cdot \sin x}{\cos ^{2}x}=\frac{\cos^{2}x+\sin^{2}x}{\cos ^{2}x}=\frac{1}{\cos ^{2}x}$

3. May 28, 2015

### phion

After the course, all of those should be second nature.

4. May 28, 2015

### Obliv

don't worry about memorizing them. As you perform integration overtime they will come second nature as stated above. Just make sure to do many practice problems each lesson/chapter

5. May 28, 2015

### mathwonk

just memorize the derivatives of sin, sec, and tangent. then the others are minus the cofunctions of those. i.e. if d/dx sec = sec.tan, then d/dx csc = -csc.cot. etc...

6. Jun 1, 2015

### David Carroll

My habit is always to go back to first principles. In the case of the derivatives of trigonometric functions, I simply memorized the derivatives of the sine and cosine. I'm not sure what Calculus textbooks you all have used, but the one I went through (Larson, 8th edition) did not give a proof that the derivative of sine is cosine, but did use this fact to establish the other derivatives and anti-derivatives. Other than this counter-example, Larson was great at showing proofs.

Do I have to wait till Advanced Calculus to find the proof that the derivative of the sine is the cosine?

7. Jun 1, 2015

### phion

You shouldn't have to, no.

8. Jun 4, 2015

### David Carroll

What am I talking bout, Arnold? I looked back to Larson's Calculus and there was such a proof. I was wrong.

9. Jun 4, 2015

### Staff: Mentor

Yes, it would very surprising for a calculus text to not show a proof of at least one trig function, using the definition of the derivative. Once you have the derivative of either sine or cosine, then you can get the derivatives of the other trig functions by the use of other techniques. IOW, if it has been proven that d/dx(sin(x)) = cos(x), then you can get d/dx(cos(x)) by noting that $cos(x) = sin(\pi/2 -x)$, and differentiating the latter using the chain rule.

10. Jun 14, 2015

### MidgetDwarf

Thomas calculus with analytic geometry 3rd ed, has a very nice proof. Can be found for 10 dollars max, and it is supperior to Larson. It also shows the trigonometric properties and how they can be derived from 2 graphs.

11. Jun 14, 2015

### Staff: Mentor

12. Jun 14, 2015

### phion

That wasn't illegally posted on the internet.

13. Jun 16, 2015

### FL0R1

Thats for sure that you have to memorize all the trigonometric integrals formula or you can simply learn the base formula and with some multiplying or substracting it goes to another formula, but if you are in a test memorizing those formulas are a good key for you.