How to Calculate the Integral of sec(x)dx: A Step-by-Step Guide

∫sec(x)dx

To calculate the integral of sec(x)dx, we can use integration by substitution

To calculate the integral of sec(x)dx, we can use integration by substitution.

Let’s define the substitution u = sec(x). Then, we need to find the derivative of u with respect to x, which is du/dx.

Since u = sec(x), we can rewrite it as u = 1/cos(x). Using the quotient rule, we can find du/dx as follows:

du/dx = [d(1)/dx * cos(x) – 1 * sin(x)] / cos^2(x)
= 0 * cos(x) – sin(x) / cos^2(x)
= -sin(x) / cos^2(x)
= -sin(x)sec^2(x)

Now, we have the substitution u = sec(x) and du/dx = -sin(x)sec^2(x).

Next, we substitute the values of u and du/dx into the original integral:

∫sec(x)dx = ∫ (u * du/dx) dx = ∫ u * (-sin(x)sec^2(x)) dx
= -∫u * sin(x)sec^2(x) dx

To simplify this further, we can use the identity sin(x)/cos(x) = tan(x) to rewrite sec^2(x) as 1 + tan^2(x):

∫sec(x)dx = -∫u * sin(x)sec^2(x) dx
= -∫u * sin(x)(1 + tan^2(x)) dx

Now, let’s rearrange the equation above to isolate dx:

dx = -du/(u * sin(x)(1 + tan^2(x)))

Substituting dx and u back into the integral, we get:

∫sec(x)dx = -∫(u * sin(x)(1 + tan^2(x))) * (-du/(u * sin(x)(1 + tan^2(x))))
= ∫du
= u + C

Finally, we need to substitute back u = sec(x) and add the constant of integration C:

∫sec(x)dx = sec(x) + C

Therefore, the integral of sec(x)dx is sec(x) + C.

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