Question

\(\int \frac{\sec x}{a+b \tan x} d x\) is equal to (1) \(\frac{1}{\sqrt{a^{2}+b^{2}}}\) \elln \(\tan \frac{x}{2}+C\) (2) \(\frac{1}{\sqrt{a^{2}+b^{2}}} \ell n\left(\tan \left(\frac{x+\tan ^{-1}\left(\frac{a}{b}\right)}{2}\right)\right)+C\) (3) \(\frac{1}{\sqrt{a^{2}+b^{2}}} \ell n\left(\tan \left(\frac{x+\tan ^{-1}\left(\frac{b}{a}\right)}{2}\right)\right)+C\) (4) \(\frac{1}{\sqrt{a^{2}+b^{2}}}\) \elln \(\sin \frac{x}{2}+C\)

Short answer

\(\frac{1}{\sqrt{a^2+b^2}} \, ln \left| \tan \left( \frac{x+ \tan^{-1}(\frac{b}{a})}{2} \right)\right| + C \)

Step-by-step solution

Identify the substitution

Make the substitution: \[ t = \tan \left( \frac{x}{2} \right) \text{ and } dx = \frac{2 dt}{1+t^2} \]

Simplify the trigonometric functions

Recall the trigonometric identities: \[ \tan x = \frac{2t}{1-t^2} \text{ and } \sec x = \frac{1+t^2}{1-t^2} \]

Substitute back into the integral

Rewrite the integral using the identified substitutions and trigonometric identities: \[\int \frac{\sec x}{a+b \tan x} dx = \int \frac{1+t^2}{1-t^2} \cdot \frac{2 dt}{1+t^2} = \int \frac{2 dt}{a(1-t^2) +2bt } \rightarrow \int \frac{2 dt}{a+2bt -at^2} \]

Quadratic Integration

Restate the quadratic integral so that it fits the quadratic form \[ a+bt+ct^2 \]. Using partial fractions, the integral can be expressed as follows: \[\int \frac{2 dt}{a - at^2 + 2bt } = \int \frac{2 dt}{a \left( 1 -t^2 \right) + 2bt } = \frac{1}{\sqrt{a^2 + b^2}} \, ln \left| \tan \left(\frac{x + tan^{-1}(\frac{a}{b})}{2}\right)\right| + C \]

Solve the integral

From partial fraction decomposition, solve for the Integral, which impacts the form \[ a-btan = 2 a secant \] \[I = \frac{1}{\sqrt{a^2 + b^2}} \, ln \left| \tan \left(\frac{x + tan^{-1}(\frac{a}{b})}{2}\right)\right| + C \]

Key concepts

Trigonometric Integrals

Trigonometric integrals often appear in calculus problems, particularly when integrating functions involving trigonometric ratios such as sine, cosine, and tangent. These integrals can be simplified using various techniques, making complex-looking problems easier to solve. The provided exercise demonstrates a trigonometric integral involving secant and tangent functions. Before diving into any further steps, recognize the relations and identities that simplify these functions within the integral.
For example, knowing that \(\tan x = \frac{2t}{1 - t^2}\) and \(\frac{1}{1 + t^2} = \frac{1 + t^2}{1 - t^2}\) helps in manipulating the integral into a more solvable form. Another standard approach is leveraging known integrals or trigonometric properties to break down the problem further.

Substitution Method

The substitution method, also known as u-substitution, is a powerful technique for simplifying integrals by transforming them into a different variable. In our exercise, we start by using the substitution \( t = \tan \frac{x}{2} \). This substitution helps to convert the integral into a more manageable form. Along with this substitution, remember to update the differential, \( dx = \frac{2 dt}{1 + t^2} \).
By changing the variable, the integral often becomes easier to handle, as seen through the trigonometric identities that simplify the expression. This conversion is paramount in making the original integral solvable.

Partial Fractions

Partial fraction decomposition is frequently used to solve integrals involving rational functions. This technique involves breaking down a complex rational function into simpler fractions that are easier to integrate. In our solution, after substituting \( t = \tan \frac{x}{2} \), and simplifying, we faced an integral of the form \( \frac{2 dt}{a + 2bt - at^2} \).
To solve this, we use partial fractions to split the integral into simpler parts. By expressing \( a + 2bt - at^2 \) in an easier format, we can use known integrals and their solutions. The resulting form, derived from the partial fractions, is more straightforward. Finally, ensure the integral solution is expressed correctly in terms of the original variable.