Problem 16 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int 4 \frac{\cos x}{\sin ^{2} x} d x$$

Step-by-Step Solution

Answer

The antiderivative of the given equation is \(-4\cot x + C\).

Step 1: Identify Substitution

Let's set a substitution such that \(u = \sin x\). This is done because in the integral, there's \(\cos x\) and \(\sin^{-2} x\), which is derivative and base of \(\sin x\) respectively. So, the equation becomes: \(\int 4 \frac{1}{u^2} du\)

Step 2: Solve the Integral

The above equation can be rewritten as \(-4 \int u^{-2} du\). Using the power rule for integrals, which states \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration, the equation turns into \(=-4u^{-1} + C\)

Step 3: Substituting Back

Substitute \(u\) back into the equation with the original value we assigned (\(u = \sin x\)). This gives us \(-4\cot x + C\)

Key Concepts

Integration by Substitution

Understanding integration by substitution is like learning a secret shortcut to solving complex integrals. Imagine you're navigating through a labyrinth of mathematical functions, Integral by substitution offers a way to simplify that path.

The method requires identifying a part of the integral that can be replaced with a single variable, making the integral simpler to solve. Typically, we look for a function within the integral whose derivative is also present. For example, if we have an integral that includes \(x^2\) and \(2x\), we might set \(u=x^2\) since the derivative of \(x^2\) is \(2x\), and \(2x\) is part of our integral.

By doing this substitution, we transform the original integral into one in terms of \(u\), which is often easier to tackle. After integrating with respect to \(u\), don't forget to substitute back to the original variable to find the final antiderivative.

Power Rule for Integration

The power rule for integration is the bread and butter of finding antiderivatives for power functions. Sandy beaches and gentle ocean waves – that's how smoothly it works for integrals involving \(x^n\).

Here's the scoop: to integrate a power function \(x^n\), you'll add one to the exponent and then divide by that new exponent, plus throw in a constant of integration for good measure. So, the integral \(\int x^n \, dx\) becomes \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is a symbol for any constant, since antiderivatives are unique up to an additive constant.

But watch out—this rule doesn't work when \(n=-1\) because it leads to division by zero! For \(x^{-1}\), or \(\frac{1}{x}\), the antiderivative is the natural logarithm of the absolute value of \(x\) plus the constant of integration.

Trigonometric Integration

Diving into trigonometric integration takes us to a world swirling with sines and cosines, where the waves of calculus ebb and flow. This technique deals with integrating functions involving trigonometric functions such as sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)), to name a few.

Common Trigonometric Integrals

Some of the fundamentals in trigonometric integration include knowing that the integral of \(\sin x\) is \(-\cos x + C\), and the integral of \(\cos x\) is \(\sin x + C\). Remember these, and you're already surfing the trigonometric wave!
When more complex trigonometric functions are involved, such as \(\sec^2 x\), you might recall that it's the derivative of \(\tan x\), which hints that its integral is \(\tan x + C\).

For the integral presented in our exercise, we saw the combination of a trigonometric function with a power, which we tackled with a substitution before applying our power rule—demonstrating how these techniques often work hand in hand.