Question

Evaluate the following integrals. $$\int \frac{dx}{1+\tan x}$$

Short answer

Question: Evaluate the integral $$\int \frac{dx}{1+\tan x}$$ Answer: $$\ln |\cos x+\sin x|+C$$

Step-by-step solution

Make a substitution

Replace $\tan x$ with $\frac{\sin x}{\cos x}$ in the given integral. $$\int \frac{dx}{1+\frac{\sin x}{\cos x}}$$

Simplify the expression

Multiply the numerator and denominator by $\cos x$. $$\int \frac{\cos x\cdot dx}{\cos x+\sin x}$$

Make another substitution

Let $u=\cos x+\sin x$. Then, $du=(-\sin x+\cos x)dx$.

Find the new integral in terms of $u$

Rewrite the original integral in terms of $u$. $$\int \frac{1}{u}du$$

Evaluate the integral

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln |u|+C$. $$\int \frac{1}{u}du=\ln |u|+C$$

Undo the substitution

Replace $u$ with $\cos x+\sin x$. $$\ln |\cos x+\sin x|+C$$ So, the final evaluated integral is: $$\int \frac{dx}{1+\tan x}=\ln |\cos x+\sin x|+C$$

Key concepts

Integration by Substitution

Integration by substitution, often referred to as u-substitution, is a technique used to simplify integrals. The idea is to substitute a portion of the integrand with a new variable, typically denoted as 'u', which turns the integral into a simpler form that is easier to evaluate. In the given exercise, the substitution was cleverly chosen as $u=\text{cos}x+\text{sin}x$, which simplifies the complex trigonometric integral. The key to successful integration by substitution is recognizing a function and its derivative within the integral.

By substituting, we transform the original integral into a form that we can integrate, often resulting in a standard integral that has a known antiderivative. This technique is very powerful and can be applied to a broad range of integrals, making it one of the first tools students should reach for when integrating.

Trigonometric Integrals

Trigonometric integrals are those that involve trigonometric functions such as \text{sin}(x)\, \text{cos}(x)\, and \text{tan}(x)\, among others. The integral in our exercise is a trigonometric integral since it involves \tan(x)\, which is a ratio of \text{sin}(x)\ and \text{cos}(x)\. To evaluate these integrals, one might use identities or algebraic manipulations to express the integral in a more convenient form.

For instance, to evaluate $\frac{dx}{1+\tan x}$ conveniently, we multiply the numerator and the denominator by \text{cos}(x)\, to eliminate the complex fraction. This step is crucial as it simplifies the expression into a form that allows us to apply the integration by substitution method mentioned earlier.

Indefinite Integrals

Indefinite integrals represent a family of functions that includes all antiderivatives of a given function. They are depicted by the integral sign without upper or lower limits and include a constant of integration (typically 'C') to account for the unknown constant in antiderivative process. In our example, the indefinite integral of $\frac{1}{u}$ with respect to 'u' is written as $\int \frac{1}{u}du=\ln |u|+C$.

This is because the derivative of $\ln |u|$ with respect to 'u' is $\frac{1}{u}$, making $\ln |u|$ the antiderivative of $\frac{1}{u}$. Hence, the solution to the original trigonometric indefinite integral is expressed in terms of the natural logarithm with an absolute value to ensure the argument is positive, followed by the constant of integration.

Integration Techniques

Integral calculus offers a plethora of techniques for evaluating integrals, with each technique serving different types of functions. In addition to integration by substitution, there are methods like integration by parts, partial fraction decomposition, and trigonometric substitution. Each method has specific scenarios where it is most effective. For example, integration by parts is frequently used when the integrand is a product of two functions, one of which becomes simpler upon differentiation.

In the exercise, integration by substitution is the chosen technique due to its suitability for the integrand's form. However, knowing when and how to apply each technique is critical for efficiently solving integrals. To master these techniques, practice is essential, as it helps in recognizing patterns and deciding the most effective strategy to tackle a given integral.