The Step-by-Step Guide to Finding the Integral of sec(x)tan(x) Using U-Substitution

integral of secxtanx

To find the integral of sec(x)tan(x), we can use a technique called u-substitution

To find the integral of sec(x)tan(x), we can use a technique called u-substitution. Let’s go through the steps:

Let’s start by writing down the integral:

∫ sec(x)tan(x) dx

Step 1: Use a substitution u = sec(x)

To do this, we differentiate both sides of the equation u = sec(x) with respect to x:

du/dx = sec(x)tan(x)

From this, we can rearrange the equation to solve for tan(x):

tan(x) = (du/dx) / sec(x)

Step 2: Replace sec(x)tan(x) and dx in the original integral

Now, we have:

∫ sec(x)tan(x) dx = ∫ (du/dx)/u dx

Step 3: Simplify the integral

Let’s rewrite the integral with respect to u:

∫ (1/u) du

Step 4: Evaluate the integral with respect to u

The integral of (1/u) du is equal to ln|u| + C, where C is the constant of integration.

Step 5: Replace u with sec(x)

Finally, substituting u back into the expression, we get:

∫ sec(x)tan(x) dx = ln|sec(x)| + C

Therefore, the integral of sec(x)tan(x) with respect to x is ln|sec(x)| + C, where C represents the constant of integration.

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