Problem 6 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 5 \sin (1-2 x) d x, \text { with } u=1-2 x $$

Step-by-Step Solution

Answer

The integral is \( \frac{5}{2} \cos(1-2x) + C \).

Step 1: Apply the Substitution

First, we substitute \( u = 1 - 2x \). This means that we need to express \( dx \) in terms of \( du \). Differentiate \( u \) with respect to \( x \): \( \frac{du}{dx} = -2 \). This gives us \( du = -2 \, dx \), so \( dx = -\frac{1}{2} \, du \).

Step 2: Substitute in the Integral

Replace \( 1-2x \) with \( u \) and \( dx \) with \( -\frac{1}{2} du \) in the integral: \( \int 5 \sin(1-2x) \, dx = \int 5 \sin(u) \left(-\frac{1}{2}\right) du \).

Step 3: Simplify the Integral

Simplify the integral: \( \int 5 \sin(u) \left(-\frac{1}{2}\right) du = -\frac{5}{2} \int \sin(u) \, du \).

Step 4: Evaluate the Integral

The integral of \( \sin(u) \) is \( -\cos(u) + C \) (where \( C \) is the constant of integration). So: \( -\frac{5}{2} \int \sin(u) \, du = -\frac{5}{2} (-\cos(u)) + C = \frac{5}{2} \cos(u) + C \).

Step 5: Substitute Back for x

Replace \( u \) back with \( 1-2x \) to express the result in terms of \( x \): \( \frac{5}{2} \cos(u) + C = \frac{5}{2} \cos(1-2x) + C \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in integration. It simplifies an integral by changing variables. This method is akin to the chain rule in differentiation. When faced with a composite function, like in the given exercise, substitution can make the problem simpler to handle.

In this problem, we substitute a part of the integral with a new variable. Specifically, we choose the inside function of the trigonometric expression:

• Let \( u = 1 - 2x \).
• Next, we need to express \( dx \) in terms of \( du \).
• Differentiating \( u \) with respect to \( x \) gives \( \frac{du}{dx} = -2 \).
• So, \( du = -2 \ dx \) or \( dx = -\frac{1}{2} \, du \).

This substitution allows us to transform the original integral into a simpler form. Integrating using \( u \) rather than \( x \) often simplifies the process significantly.

Integration Techniques

Once substitution transforms the integral, the next step is to simplify the integration process. Simplification involves replacing variables and adjusting constants.

• The original integral \( \int 5 \sin(1-2x) \, dx \) becomes \( \int 5 \sin(u) \left(-\frac{1}{2}\right) du \) after substitution.
• This simplification results in \( -\frac{5}{2} \int \sin(u) \, du \).
• The constant \(-\frac{5}{2}\) can be factored out of the integral. This follows the properties of integrals, allowing constants to be removed for easier calculation.

Thereafter, the integration task becomes straightforward, focusing only on \( \sin(u) \). This streamlined process showcases how effective integration techniques are in handling complex integrals.

Trigonometric Integrals

Trigonometric integrals involve integrating functions of sine, cosine, and other trigonometric functions. In our case, we deal specifically with the sine function.

• Trigonometric integrals require standard integration formulas to solve effectively.
• The integral of \( \sin(u) \) is \( -\cos(u) + C \), where \( C \) is the constant of integration.
• In our example, \( -\frac{5}{2} \int \sin(u) \, du \) integrates to \( \frac{5}{2} \cos(u) + C \).

Ultimately, trigonometric integrals require a good grasp of trigonometric identities and their integrals. Once the integration is complete, it's critical to substitute back the original variable to express the answer correctly. In this exercise, substituting back \( u = 1-2x \) gives our final answer: \( \frac{5}{2} \cos(1-2x) + C \). This solution embodies how trigonometric integrals are approached systematically with precision.