Mastering Integration by Substitution | Solving the Integral of sec(x) with Respect to x

∫(secx)dx

To solve the integral of sec(x) with respect to x, you can use a technique called integration by substitution

To solve the integral of sec(x) with respect to x, you can use a technique called integration by substitution. Here’s how you can do it:

Step 1: Begin by rewriting sec(x) as 1/cos(x).

∫(sec(x)) dx = ∫(1/cos(x)) dx

Step 2: Let u = cos(x). Differentiate both sides of this equation with respect to x to find du:

du/dx = -sin(x)

Step 3: Rearrange the equation to solve for dx:

dx = du / (-sin(x))

Step 4: Substitute the u and dx values from steps 2 and 3 into the integral:

∫(1/cos(x)) dx = ∫(1/u) (du / -sin(x))

Step 5: Rewrite the integral in terms of u:

∫(1/u) (du / -sin(x)) = -∫(1/u) (du / sin(x))

Step 6: Simplify the integral:

-∫(1/u) (du / sin(x)) = -∫(du / (u * sin(x)))

Step 7: Recognize that the new integral can be solved using a natural logarithm:

-∫(du / (u * sin(x))) = -ln|u| + C

Step 8: Substitute the value of u from step 2:

-∫(du / (u * sin(x))) = -ln|cos(x)| + C

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

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