Question

Evaluate the following integrals. $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x$$

Short answer

Question: Evaluate the integral $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x$$ Answer: $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x = - \frac{1}{4} (\tan x)^{-4} + C$$

Step-by-step solution

Choose the substitution

To simplify the expression, let's make a substitution. We can let $$u = \tan x$$ Then $$\frac{du}{dx} = \sec^2 x$$ So, $$dx = \frac{du}{\sec^2 x}$$ Now we'll rewrite the integral using the substitution 'u':

Rewrite the integral with the substitution

Substitute the expressions in terms of 'u' into the given integral: $$\int \frac{\sec^2 x}{\tan^5 x} dx = \int \frac{1}{u^5} \frac{du}{\sec^2 x}$$

Evaluate the new integral

From Step 1, we have seen that: $$dx = \frac{du}{\sec^2 x}$$ And so, $$\int \frac{1}{u^5} \frac{du}{\sec^2 x} = \int u^{-5} du$$ Now, we can find the antiderivative using the power rule: $$\int u^{-5} du = - \frac{1}{4} u^{-4} + C$$

Substitute back using 'x'

Finally, we need to substitute back in terms of 'x' using the original substitution: $$- \frac{1}{4} u^{-4} + C = - \frac{1}{4} (\tan x)^{-4} + C$$ So, the final answer is: $$\int \frac{\sec^2 x}{\tan^5 x} dx = - \frac{1}{4} (\tan x)^{-4} + C$$

Key concepts

Integration by Substitution

Integration by substitution is a technique used to simplify complex integrals. It involves replacing a difficult-to-integrate expression with a simpler one by introducing a new variable. This technique can be considered as the reverse process of the chain rule in differentiation.

The effectiveness of substitution hinges on the ability to recognize a part of the integrand that can be substituted with a new variable, often resulting in an integral in terms of standard forms that are easier to evaluate. For instance, in the exercise \( \int \frac{\sec^2 x}{\tan^5 x} dx \), the substitution \( u = \tan x \) simplifies the integral by turning it into an expression with a power of the new variable \( u \), thus making it straightforward to integrate using basic integration rules.

To correctly perform the substitution, one must also express the differential \( dx \) in terms of the new variable, as done in \( dx = \frac{du}{\sec^2 x} \). By replacing the variables and differentials in the original integral, the new integral in terms of \( u \) becomes easier to handle and paves the way for applying further integration rules.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function that we are integrating. In other words, the antiderivative of a function \( f(x) \) is another function \( F(x) \) such that \( F'(x) = f(x) \).

The process of finding an antiderivative is crucial in solving an integral, as it essentially reverses the operation of differentiation. When we integrate a function, we are looking for all possible antiderivatives of that function. These are represented by the general form of the antiderivative plus a constant of integration, \( C \), since differentiation of a constant yields zero.

In the given exercise, finding the antiderivative involves determining the function whose derivative is \( u^{-5} \). The result is \( -\frac{1}{4} u^{-4} + C \), where \( C \) is the constant of integration. The antiderivative represents the family of all functions that, when differentiated, would give us the original function \( u^{-5} \).

Power Rule for Integration

The power rule for integration is a fundamental technique used to find the antiderivative of a function of the form \( x^n \), where \( n \) is any real number except -1. According to this rule, to integrate a power function, we increase the exponent by one and divide by the new exponent, and then add the constant of integration.

Formally, the power rule for integration is expressed as:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \] for all \( n \eq -1 \).

Applying the power rule to the exercise's substitution-adjusted function \( u^{-5} \), we increase the exponent by one from -5 to -4 and then divide by -4, obtaining the antiderivative \( -\frac{1}{4} u^{-4} \). After integrating, we add the constant \( C \) to account for all possible antiderivatives. This rule simplifies the process of integration by providing a direct method to integrate power functions, allowing us to find the solution quickly and efficiently.