Solve Trig Integral: x^5 + x^5 + [cos(x)]^6

[lovemake1]

Homework Statement

Integral of ...
h(x) = x^5 + x^5 + [cos(x)]^6

Homework Equations

The Attempt at a Solution

so it would be

1/6x^6 + 1/6x^6 + 1/7[sin(x)]^7
is this correct?

 

[Simon Bridge]

You offer: $$\int \left ( 2x^5 + \cos^6(x) \right )dx = \frac{1}{3}x^6 + \frac{1}{7}\sin^7(x)+c$$ That would not be correct - look up how to integrate a power of a trig function.
One way of checking your integration is to differentiate it and see if you get the function you started with.

 

[lurflurf]

hint:
cos(x)^6=(1/32)(15cos(2x)+6cos(4x)+cos(6x)+10)

 

[lovemake1]

wait is there diference between
cos(x)^6 and [cos(x)]^6

 

[dextercioby]

Hmm, you should use proper bracketing, if unable to use exponents: cos⁶ x is from putting the 6 between the [sup ] and [/ sup] (without the 2 spaces).
I write cos x without the () next to x. It looks nicer to me.

 

[Mark44]

wait is there diference between
cos(x)^6 and [cos(x)]^6

Probably not, but you won't see the first form you wrote very often.

These mean the same:

[cos(x)]⁶ and cos⁶(x)

For each of these, you take the cosine of a number x, and then raise that result to the 6th power.

Both of the above are different from this:

cos(x⁶)

For this one, you raise x to the 6th power, and then take the cosine of that result.

 

[Mark44]

Integral of ...
h(x) = x^5 + x^5 + [cos(x)]^6

Do you have a typo? Why is x⁵ written twice? If it's not a mistake, h(x) should be simplified to h(x) = 2x⁵ + cos⁶(x)

 

[Simon Bridge]

@lovemake1: do you know how lurflurf knew that cos^6(x)=(1/32)(15cos(2x)+6cos(4x)+cos(6x)+10) ? You don't go around memorizing these things (well: you can if you want) and you don't need to look it up anywhere.

(tldr: use LaTeX)
Typsetting is important for communication - especially when it comes to math: you have take care when representing math in plain text - you can easily end up writing something ambiguous. i.e. you noticed that "cos(x)^6" is a bit ambiguous whether the power applies to the x or the cosine of x... however you wrote "1/6x^6" which is a bit ambiguous whether the "x^6" is in the numerator or the denominator. In each case the respective author is relying on the context to make the matter clear.

A few style guides allow leaving off the parentheses (as per dextercioby's suggestion - note: he uses a space after the trig?) after the trig function so you can write sinx instead of sin(x) but the latter is clearer in text (with proper typesetting, the "sin" part is upright and the "x" part is italic, like this: $\sin x$, to make it clearer) ... but then sinxy is a bit unclear if it is sin(x)y or sin(xy) ... to make the distinction, you have to premultiply trig functions so you'd write ysinx for the first version ... or, better, use dots to distinguish the groupings: y.sinx is better than ysinx is better than sinx.y but would be understood to mean the same. This also allows for trig functions to be multiplied together as in sinx.cosy which is not the same as sin(x.cosy). when you nest trig function like that last one, you need to be explicit with the brackets or (rarely done) use the "follows" notation so "sin o x.cosy = sin(x.cosy)". Most feel the latter is better.

So plain text is a pain in the proverbial. PF provides a gui equation builder in the advanced editor but experienced members prefer to type the markup in directly. You get a choice of bv style markup or a LaTeX environment if you feel that more clarity is warranted.

Practically everyone uses the LaTeX in the end - it is needed at college level, and it is really good to learn it earlier than you really need it.