Solving the Integral of sec(x) dx using u-substitution: Step-by-Step Guide

secx dx

To integrate sec(x) dx, we can use a technique called u-substitution

To integrate sec(x) dx, we can use a technique called u-substitution. Let’s go step by step:

Step 1: Identify the function to be substituted.
In this case, the function to be substituted is sec(x).

Step 2: Determine the substitution.
Let’s make the substitution u = tan(x). We can rewrite sec(x) in terms of tan(x) as:
sec(x) = 1/cos(x) = sqrt(1 + tan^2(x)) = sqrt(1 + u^2)

Step 3: Calculate the differential of u.
To determine the differential of u, we differentiate u = tan(x) with respect to x:
du/dx = sec^2(x)

Step 4: Solve for dx.
Rearranging the equation du/dx = sec^2(x), we get dx = du/sec^2(x)

Step 5: Substitute the variables and integrate.
Now we substitute the values from steps 2 and 4 into the integral:
∫ sec(x) dx = ∫ 1/cos(x) dx = ∫ 1/(sqrt(1 + tan^2(x))) dx

Substituting in the values from step 4, we have dx = du/sec^2(x), so the integral becomes:
∫ 1/(sqrt(1 + u^2)) * (du/sec^2(x))

To simplify further, we can replace sec^2(x) with (1 + tan^2(x)):
∫ 1/(sqrt(1 + u^2)) * (du/(1 + tan^2(x)))

Now, our integral becomes:
∫ 1/(sqrt(1 + u^2)) * (du/(1 + u^2))

At this point, we can simplify the integral to:
∫ du/sqrt(1 + u^2)

Step 6: Solve the integral.
This integral is a well-known trigonometric integral. We can solve it by using a trigonometric substitution.

Let u = sin(theta), then du = cos(theta) d(theta).
Substituting these values, our integral becomes:
∫ (cos(theta) d(theta))/(sqrt(1 + sin^2(theta)))

We know that 1 + sin^2(theta) = cos^2(theta), so we can simplify the integral further:
∫ (cos(theta) d(theta))/cos(theta)
= ∫ d(theta)

Integrating d(theta) gives:
θ + C

Step 7: Substitute back the original variable.
Since we made the substitution u = tan(x), we can replace θ with tan^(-1)(u):
θ + C = tan^(-1)(u) + C

Final step: Substitute back u = tan(x) into the expression.
Therefore, the final answer is:
∫ sec(x) dx = tan^(-1)(u) + C
= tan^(-1)(tan(x)) + C

Thus, the integral of sec(x) dx is tan^(-1)(tan(x)) + C, where C is the constant of integration.

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