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How do you integrate #cos(5x)dx#?

Answer 1

Christopher Allers

Let #u = 5x#. Then #du = 5dx# and #dx= (du)/5#.

#I = int 1/5cos(u) du#

#I = 1/5sin(u) + C#

#I = 1/5sin(5x) + C#

Hopefully this helps!

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Answer 2

Emily Ammerman

To integrate ( \cos(5x) , dx ), you can use the substitution method. Let ( u = 5x ), then ( du = 5 , dx ). Rearranging, we have ( dx = \frac{du}{5} ). Substituting into the integral:

[ \int \cos(5x) , dx = \int \cos(u) \frac{du}{5} = \frac{1}{5} \int \cos(u) , du ]

Now, integrate ( \cos(u) ) with respect to ( u ):

[ \frac{1}{5} \int \cos(u) , du = \frac{1}{5} \sin(u) + C ]

Substitute back ( u = 5x ):

[ \frac{1}{5} \sin(5x) + C ]

So, the integral of ( \cos(5x) ) with respect to ( x ) is ( \frac{1}{5} \sin(5x) + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.