Problem 68 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \csc ^{2}(2 x) d x $$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{2} \cot(2x) + C\).

Step 1: Identify the Appropriate Integration Formula

The integral presented is \( \int \csc^2(2x) \, dx \). We recognize that the integral of \( \csc^2(u) \) with respect to \( u \) is \( -\cot(u) + C \). However, we have \( 2x \) instead of just \( x \), so we need to account for this in our integration process.

Step 2: Use Substitution

To solve this integral, we need to perform a substitution. Let \( u = 2x \). Then the derivative \( du = 2 \, dx \) or rearranging, \( dx = \frac{1}{2} du \). Substitute \( u \) and \( dx \) into the integral: \[ \int \csc^2(u) \cdot \frac{1}{2} \, du \].

Step 3: Integrate Using the Known Formula

Now that the integral is in a familiar form, \( \int \frac{1}{2} \csc^2(u) \, du \), you can use the antiderivative \( \int \csc^2(u) \, du = -\cot(u) + C \). Therefore, the integral becomes \[ \frac{1}{2} (-\cot(u)) + C = -\frac{1}{2} \cot(u) + C \].

Step 4: Substitute Back the Original Variable

To return to the variable \( x \), substitute back \( u = 2x \). This gives us \[ -\frac{1}{2} \cot(2x) + C \] as our indefinite integral of the original function.

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique for finding antiderivatives. It's especially useful when you're dealing with more complex integrals where the variable of integration is not as straightforward. The key idea is to simplify the integral into a form that is easier to integrate by substituting part of the expression with a single variable.

To use substitution, you start by identifying a part of the integrand that will simplify the integration process. In our example, we set \( u = 2x \). Then, you differentiate this expression to find \( du \). Hence, \( du = 2 \, dx \) which implies \( dx = \frac{1}{2} \, du \).

Substitution transforms the original integral into an easier one. After performing the integral, you switch back to your original variable. This method allows you to analyze and solve problems that initially seem challenging. It’s like turning a hard puzzle into an easier one with the help of a translator.

Csc Function Integration

The cosecant function, \( \csc(x) \), is the reciprocal of the sine function. Its properties make it particularly interesting and sometimes tricky to integrate directly.

When integrating \( \csc^2(x) \), the corresponding integral formula is \( \int \csc^2(x) \, dx = -\cot(x) + C \). The cotangent function, \( \cot(x) \), is the reciprocal of the tangent function, adding another layer of depth when dealing with trigonometric integrals.

In the given example, the integration doesn’t occur with \( x \) alone, but rather \( 2x \). This is where understanding substitution becomes crucial, as we substitute \( u = 2x \) to employ the known integration formula for \( \csc^2(u) \). Integrating trigonometric functions like \( \csc \) can often involve recognizing these familiar patterns and using substitution to navigate more complex scenarios.

Antiderivatives

An antiderivative of a function is essentially the reverse of differentiation. When you find an antiderivative, you’re determining the function whose derivative gives you the original function back.

The process allows us to determine indefinite integrals, where the constant of integration \( C \) is always added because differentiation of a constant results in zero. Thus, the integral outcomes aren't unique unless an additional condition is provided. Implementing this in our example, we identify that \( \int \csc^2(u) \, du = -\cot(u) + C \).

After finding the antiderivative in terms of the substituted variable, the final step is substituting back to express everything in terms of the original variable. This gives the complete solution as \( -\frac{1}{2} \cot(2x) + C \). Recognizing and correctly implementing antiderivatives help unlock the solutions to indefinite integrals.