Integration by Substitution | Evaluating the Integral of sin(ax) with Respect to x

∫ sin(ax) dx =

To evaluate the integral of sin(ax) with respect to x, we can use a technique called integration by substitution

To evaluate the integral of sin(ax) with respect to x, we can use a technique called integration by substitution.

Let’s start by making a substitution. We can let u = ax. Then, we need to find the differential of u with respect to x, which is du/dx = a.

Rearranging this equation, we get dx = du/a.

Substituting these values back into the integral, we have:

∫ sin(ax) dx = ∫ sin(u) (du/a)

Now, the integral has been transformed into an integral of sin(u) with respect to u, which we can evaluate easily. The integral of sin(u) is -cos(u) + C, where C is the constant of integration.

Therefore, the final answer to the integral is:

∫ sin(ax) dx = -cos(u) + C

But we need to substitute back the original variable u = ax:

= -cos(ax) + C

So, the integral of sin(ax) with respect to x is -cos(ax) + C, where C is the constant of integration.

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