Problem 20 - Applied Calculus

Question

Evaluate the integrals in both exactly [e.g. \(\ln (3 \pi)] \text { and numerically [e.g. } \ln (3 \pi) \approx 2.243]\). $$\int_{3}^{5} x \cos x d x$$

Step-by-Step Solution

Answer

The integral from 3 to 5 of \( x \cos x \) is approximately \(-6.435\).

Step 1: Integration by Parts Formula

The integral involves the product of a polynomial and a trigonometric function. For such integrals, integration by parts is often useful, where \( \int u \, dv = uv - \int v \, du \). Choose \( u = x \) and \( dv = \cos x \, dx \).

Step 2: Differentiate and Integrate Parts

Differentiate \( u = x \), giving \( du = dx \); integrate \( dv = \cos x \, dx \), resulting in \( v = \sin x \).

Step 3: Apply Integration by Parts

Substitute into the integration by parts formula: \( \int x \cos x \, dx = x \sin x - \int \sin x \, dx \).

Step 4: Solve Remaining Integral

Now solve \( \int \sin x \, dx \). The integral of \( \sin x \) is \( -\cos x \). Thus, we have \( \int x \cos x \, dx = x \sin x + \cos x + C \).

Step 5: Evaluate Definite Integral

Evaluate the definite integral from 3 to 5: \[ \int_{3}^{5} x \cos x \, dx = \left[x \sin x + \cos x\right]_3^5. \]

Step 6: Evaluate at Upper Limit

Calculate at \( x = 5 \): \( 5 \sin(5) + \cos(5) \).

Step 7: Evaluate at Lower Limit

Calculate at \( x = 3 \): \( 3 \sin(3) + \cos(3) \).

Step 8: Compute Difference

Subtract the lower limit result from the upper limit result: \( (5 \sin(5) + \cos(5)) - (3 \sin(3) + \cos(3)) \).

Step 9: Numerical Evaluation

Use a calculator to find approximate values: \( \sin(5) \approx -0.95892 \), \( \cos(5) \approx 0.28366 \), \( \sin(3) \approx 0.14112 \), \( \cos(3) \approx -0.98999 \). Compute the numerical result: \((5(-0.95892) + 0.28366) - (3(0.14112) - 0.98999) \approx -6.435 \).

Key Concepts

Integration by Parts

When faced with an integral involving the product of two distinct types of functions, like polynomials and trigonometric functions, "integration by parts" is a helpful technique to use. This method is akin to the product rule used in differentiation, but in reverse.

To apply this technique, we choose two parts from the integral:

• Function "u", which we differentiate
• Function "dv", for which we find the antiderivative

This results in the formula:\[ \int u \, dv = uv - \int v \, du \]For example, suppose we have an integral \( \int x \cos x \, dx \). Here:

• Let \( u = x \) and \( dv = \cos x \, dx \)
• This gives \( du = dx \) and \( v = \sin x \)

Substituting these into the formula gives:\[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx \]From here, we integrate \( \sin x \, dx \) to get \( -\cos x \), leading to the result \( x \sin x + \cos x + C \), with \( C \) being the constant of integration when handling indefinite integrals.

Definite Integral

A definite integral helps us find the area under a curve over a specific interval. Instead of yielding a general function, it produces a number by evaluating the antiderivative at the interval's endpoints.

In the process for \( \int_{3}^{5} x \cos x \, dx \), once the indefinite form \( x \sin x + \cos x \) was found using integration by parts, the next step is:

• Evaluate at the upper limit: \( x = 5 \)
• Evaluate at the lower limit: \( x = 3 \)

Substituting these into the expression, we calculate:

\[ \left[ x \sin x + \cos x \right]_3^5 \]At \( x = 5 \): \( 5 \sin(5) + \cos(5) \)
At \( x = 3 \): \( 3 \sin(3) + \cos(3) \)
The difference between these two results gives the exact value of the definite integral.

This technique is fundamental because it directly helps determine quantities such as areas, volumes, and other values integral to calculus.

Numerical Evaluation

Sometimes, the exact values for certain trigonometric functions are unwieldy, and numerical evaluations can provide a practical approximation. When evaluating expressions like \( \cos(5) \) or \( \sin(3) \), it's common to need estimates rather than exact values.

Calculators or numerical software become helpful here. For example:

• \( \sin(5) \approx -0.95892 \)
• \( \cos(5) \approx 0.28366 \)
• \( \sin(3) \approx 0.14112 \)
• \( \cos(3) \approx -0.98999 \)

Use these decimal forms to compute the difference:

\[(5(-0.95892) + 0.28366) - (3(0.14112) + (-0.98999)) \approx -6.435 \]This approximation reflects the practical impact of definite integrals, providing clearer insight into the integral's measurement in real-world contexts.