# Thread: How do you integrate the following trigonometric functions ?

1. ## How do you integrate the following trigonometric functions ?

In my previous posts people had problems understanding my question so this time I will keep it simple and objective.How do you integrate these:

• cos(5x)
• cos(5x2)
• cos(5x2+2x3)
• cos(5x2.5)
• cos(e2x)
• cos(ln[2x])
• cos(ln[x2] )

And please also tell me if there is a general method to solve all of these.

2. ## Re: How do you integrate the following trigonometric functions ?

I'm confused. Are you not taking (or have taken) a Calculus course? Generally speaking, you learn to use "substitution" in integration well before integrating trig functions. So the first thing you should think of when try to integrate $\int cos(5x)dx$ is the substitution u= 5x so that du= 5 dx so that dx= (1/5)du. The integral becomes [tex]\int cos(u)(1/5)du[tex]. Because that "1/5" is a constant, we can take it outside the integral to get $\frac{1}{5}\int cos(u)du= \frac{1}{5}sin(u)+C= \frac{1}{5}sin(5x)+ C$.

To integrate cos(5x^2), let u= 5x^2. Then du= 10xdx. And there's the rub! dx= 1/(10x) du so this would become $\int\frac{1}{10x} cos(u)du$. While we can take the constant, 1/10, out of the integral, we cannot take out the "1/x". IF the problem were $\int x cos(5x^2)dx$ THEN we could let $u= 5x^2$ so that du= 10xdx and xdx= (1/10)du so that the integral becomes $\frac{1}{10}\int cos(u)du= \frac{1}{10}sin(u)+ C= \frac{1}{10}sin(5x^2)+ C$. But without that "x" in the original integral, we cannot do that.

NO, there is no "general method to solve all of these". In fact, except for the very first, $cos(5x)$, none of the integrals can be written in terms of "elementary functions".

3. ## Re: How do you integrate the following trigonometric functions ?

Originally Posted by reindeer7
In my previous posts people had problems understanding my question so this time I will keep it simple and objective.How do you integrate these:
• cos(5x)
• cos(5x2)
• cos(5x2+2x3)
• cos(5x2.5)
• cos(e2x)
• cos(ln[2x])
• cos(ln[x2] )

And please also tell me if there is a general method to solve all of these.
This is almost the same answer that I gave you yesterday.
You can use this webpage.

If you change the input function to $\int {\cos (5x)dx}$ you sill see it is very easy to do.

Where as $\int {\cos (x^2)dx}$ looks simple but it is actually very complicated. It requires special functions to answer.

You can use that page to do each of those function in you list. You will find that only the first and last two have relatively simple solutions.