Problem 25 - Calculus for Biology and Medicine

Question

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 2} e^{x} \sin x d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2} (e^{\pi/2} + 1) \).

Step 1: Identify Parts for Integration by Parts Formula

Integration by parts formula is \( \int u \, dv = uv - \int v \, du \). We need to choose \( u \) and \( dv \) from \( e^x \sin x \, dx \). Choose \( u = \sin x \) and \( dv = e^x \, dx \).

Step 2: Calculate Derivative and Integral

Find \( du \) and \( v \):- \( du = \cos x \, dx \) (since the derivative of \( \sin x \) is \( \cos x \)),- \( v = e^x \) (since the integral of \( e^x \, dx \) is \( e^x \)).

Step 3: Apply Integration by Parts

Substitute into the formula: \(\int e^{x} \sin x \, dx = \sin x \cdot e^x - \int e^x \cos x \, dx\).

Step 4: Repeat Integration by Parts

Now, evaluate \( \int e^x \cos x \, dx \) using the same method. Let \( u = \cos x \), \( dv = e^x \, dx \). Then, \( du = -\sin x \, dx \), \( v = e^x \). Apply integration by parts again: \(\int e^x \cos x \, dx = \cos x \cdot e^x + \int e^x \sin x \, dx\).

Step 5: Combine and Solve Equation

Now we have two equations:1. \( \int e^{x} \sin x \, dx = e^x \sin x - \int e^x \cos x \, dx \).2. \( \int e^x \cos x \, dx = e^x \cos x + \int e^x \sin x \, dx \).Substitute equation 2 into equation 1, \(\int e^x \sin x \, dx = e^x \sin x - \left(e^x \cos x + \int e^x \sin x \, dx\right)\).

Step 6: Simplify and Solve

Simplify to find \(2\int e^x \sin x \, dx = e^x(\sin x - \cos x)\), and thus\(\int e^x \sin x \, dx = \frac{1}{2} e^x(\sin x - \cos x) + C\).

Step 7: Evaluate Definite Integral

Evaluate from 0 to \( \pi/2 \):\(\left[\frac{1}{2} e^x(\sin x - \cos x)\right]_0^{\pi/2}= \frac{1}{2} \left(e^{\pi/2}(1-0) - e^0(0-1)\right) = \frac{1}{2} (e^{\pi/2} + 1)\).

Key Concepts

Definite Integral

A definite integral is a way to calculate the area under a curve within a specific range. Here, the range is given by the interval \([0, \frac{\pi}{2}]\). Unlike an indefinite integral, which represents a family of functions, a definite integral provides a precise numerical result.

When dealing with a definite integral, it’s crucial to recognize the limits of integration, denoted here as the lower limit \(0\) and the upper limit \(\frac{\pi}{2}\). These limits indicate the "start" and "end" points of our calculation.

• The expression inside the integral symbol, like \(e^x \sin x\), is called the integrand.
• The process of integration accumulates the area under the curve defined by this integrand over the specified limits.

To solve, we often use different techniques such as substitution or, as in this example, integration by parts. Once the integral is evaluated, the definite integral is calculated by substituting the limit values into the resulting antiderivative and subtracting the results.

Trigonometric Functions

Trigonometric functions, like \(\sin x\) and \(\cos x\), describe relationships in a right triangle but are also essential in periodic phenomena modeling. In the integral \(\int_{0}^{\pi / 2} e^{x} \sin x dx\), the sine function cycles through its values from 0 to 1.

Understanding the basic derivatives and integrals of trigonometric functions is vital:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).
• Integrating \(\sin x\) requires recognition and rearrangement, as these functions combine in complicated expressions.

These functions provide oscillating values, often catching students by surprise when combined with other types, like exponential in this instance. Carefully track these transformations while using techniques like integration by parts to find the correct solution for integrals involving trigonometric functions.

Exponential Functions

Exponential functions appear in many forms in calculus; one common form is \(e^x\). This particular function is known for its unique property: its derivative is itself, \(d/dx(e^x) = e^x\). This property makes \(e^x\) especially manageable in integration since

• The integral of \(e^x dx\) returns \(e^x + C\).

When combined with other functions, such as in integration by parts, \(e^x\)'s consistent behavior can simplify the process:

• It doesn't change form when you differentiate or integrate, providing predictable results.
• The combination with trigonometric functions like \(\sin x\) or \(\cos x\) often requires repeated application of methods like integration by parts.

In the given problem, pairing \(e^x\) with these sinusoidal functions in the definite integral leads to expressions with solutions such as \(\frac{1}{2}(e^{\pi/2} + 1)\), leveraging the stable nature of the exponential when computed over specific bounds.