How to Find the Integral of cos(x) using Integration by Substitution

∫ cosx dx

To find the integral ∫ cos(x) dx, we can use the integration technique called integration by substitution

To find the integral ∫ cos(x) dx, we can use the integration technique called integration by substitution.

Step 1: Choose a suitable substitution.
Let’s choose u = sin(x). Note that this choice is suitable because the differential of sin(x) is cos(x) dx, which is present in the integral.

Step 2: Find dx in terms of du.
Taking the derivative of both sides of the equation u = sin(x) with respect to x, we get du/dx = cos(x).
Rearranging this equation, dx = du / cos(x).

Step 3: Rewrite the integral in terms of u.
Substituting the values from step 2 into the original integral, we get:
∫ cos(x) dx = ∫ (du / cos(x))
Since u = sin(x), we can rewrite cos(x) as √(1 – u^2) using the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

∫ (du / cos(x)) = ∫ (du / √(1 – u^2))

Step 4: Evaluate the integral in terms of u.
The expression we obtained in step 3 is now in a form that we can integrate using standard integration techniques.

The integral of 1 divided by the square root of (1 – u^2) is arcsin(u) + C, where C is the constant of integration.

So, the integral of cos(x) dx is arcsin(u) + C.

Step 5: Substitute back in terms of x.
Recall that u = sin(x). Therefore, we replace u with sin(x) in our result:

∫ cos(x) dx = arcsin(sin(x)) + C.
However, it’s important to note that this solution is not the only answer. The integral of cos(x) has a periodic nature, so we need to include the general solution. We know that sin(x) has a special property:
arcsin(sin(x)) = x + 2kπ, where k is an integer representing the number of full cycles of the sine function.

Thus, the complete solution is:
∫ cos(x) dx = x + 2kπ + C, where k is an integer, and C is the constant of integration.

Therefore, the integral ∫ cos(x) dx evaluated with respect to x is equal to x + 2kπ + C.

More Answers: