∫ cos(ax) dx =
To solve the integral ∫ cos(ax) dx, we can use integration by substitution or the trigonometric identities
To solve the integral ∫ cos(ax) dx, we can use integration by substitution or the trigonometric identities.
Method 1: Integration by Substitution
Let’s start by substituting u = ax. Taking the derivative of both sides, we get du = a dx. Rearranging this equation, we find dx = du/a.
Now, substitute these values into the integral:
∫ cos(ax) dx = ∫ cos(u) (du/a)
Since cos(u) does not depend on x anymore, we can treat it as a constant and take it out of the integral:
= (1/a) ∫ cos(u) du
Integrating cos(u) with respect to u gives us sin(u):
= (1/a) sin(u) + C
Finally, substitute u back in terms of x:
= (1/a) sin(ax) + C
So, the solution to ∫ cos(ax) dx is (1/a) sin(ax) + C, where C is the constant of integration.
Method 2: Trigonometric Identity
Alternatively, we can use the trigonometric identity cos(x) = (1/2)(e^(ix) + e^(-ix)) to solve the integral.
Substituting this identity into the original integral, we have:
∫ cos(ax) dx = ∫ (1/2)(e^(iax) + e^(-iax)) dx
Distribute the (1/2) term:
= (1/2) ∫ (e^(iax) + e^(-iax)) dx
Now we can integrate term by term:
= (1/2) ( ∫ e^(iax) dx + ∫ e^(-iax) dx )
To integrate e^(iax), we can use the substitution u = ax, which leads to du = a dx:
= (1/2a) ∫ e^(iu) du
Integrating e^(iu) gives us (1/i) e^(iu):
= (1/2ai) e^(iu) + C1
Using the same process for the second term, we get:
= (1/2ai) e^(-iu) + C2
Now, substitute back u = ax:
= (1/2ai) e^(iax) + (1/2ai) e^(-iax) + C
Combine the terms, and since (1/2ai) is a constant, we can bring it outside the integral:
= (1/2a) (e^(iax) + e^(-iax)) + C
Remembering that cos(ax) = (1/2)(e^(iax) + e^(-iax)), we can simplify further:
= (1/2a) (2cos(ax)) + C
Finally, cancel out the 2 and rewrite as:
= (1/a) cos(ax) + C
So, the solution to ∫ cos(ax) dx is (1/a) cos(ax) + C, where C is the constant of integration.
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