Finding the Integral of Sec(x) with the Method of Substitution: Step-by-Step Guide and Solution

∫secxdx

To find the integral of sec(x) with respect to x, we can use the method of substitution

To find the integral of sec(x) with respect to x, we can use the method of substitution.

Let’s assume u = tan(x), then du/dx = sec^2(x).

To express sec(x) in terms of u, we can use the Pythagorean identity: sec^2(x) = 1 + tan^2(x).
Rearranging the terms, we get tan^2(x) = sec^2(x) – 1.

Now, replace sec^2(x) in the derivative expression with u^2 – 1, so du/dx = u^2 – 1.
Rearranging, we get du = (u^2 – 1) dx.

Dividing both sides by (u^2 – 1), we have du/(u^2 – 1) = dx.
Now we can substitute these new expressions into our integral.

∫sec(x) dx = ∫du/(u^2 – 1)

Next, we need to solve the integral of 1/(u^2 – 1) with respect to u.

To do this, we can use partial fraction decomposition.
First, factor the denominator of the integral as (u – 1)(u + 1).

Now we need to find the constants A and B, such that 1/(u^2 – 1) = A/(u – 1) + B/(u + 1).
Multiply both sides of this equation by (u^2 – 1), we have:

1 = A(u + 1) + B(u – 1)
1 = Au + A + Bu – B

We set the coefficients of matching powers of u equal to each other, so we have the following equations:

A + B = 0 (coefficients of u^0)
A – B = 1 (coefficients of u^1)

Solving these equations gives us A = B = 1/2.

Now we can rewrite the integral as follows:

∫du/(u^2 – 1) = 1/2 ∫du/(u – 1) + 1/2 ∫du/(u + 1)

Using the basic integral formulas, we can solve these two integrals:

1/2 ∫du/(u – 1) = 1/2 ln|u – 1| + C1
1/2 ∫du/(u + 1) = 1/2 ln|u + 1| + C2

Here C1 and C2 are constants of integration.

Finally, we can substitute back u = tan(x) to obtain the final result:

∫sec(x) dx = 1/2 ln|tan(x) – 1| + C1 + 1/2 ln|tan(x) + 1| + C2

Combining the constants, we can write it as:

∫sec(x) dx = 1/2 ln|tan(x) – 1| + 1/2 ln|tan(x) + 1| + C

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