Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int_{0}^{\pi / 4} \frac{\cos x}{1+\sin ^{2} x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{8} \).

Step 1: Identify the Integration Technique

The given integral is \( \int_{0}^{\pi / 4} \frac{\cos x}{1+\sin ^{2} x} \, dx \). Notice the presence of both \( \cos x \) and \( \sin x \) terms. This suggests that a trigonometric substitution or manipulation could simplify the integration.

Step 2: Simplify Using Substitution

Use the substitution \( u = \sin x \). Then \( du = \cos x \, dx \). As \( x \) changes from 0 to \( \pi/4 \), \( u \) changes from \( \sin(0) = 0 \) to \( \sin(\pi/4) = \frac{\sqrt{2}}{2} \). The integral becomes: \( \int_{0}^{\frac{\sqrt{2}}{2}} \frac{1}{1+u^2} \, du \).

Step 3: Integrate the Simplified Expression

The integral \( \int \frac{1}{1+u^2} \, du \) is a standard form that evaluates to \( \arctan(u) + C \). Hence, \( \int_{0}^{\frac{\sqrt{2}}{2}} \frac{1}{1+u^2} \, du = [\arctan(u)]_{0}^{\frac{\sqrt{2}}{2}} \).

Step 4: Evaluate the Definite Integral

Now, substitute the limits back: \( \arctan\left(\frac{\sqrt{2}}{2}\right) - \arctan(0) \). Knowing that \( \arctan(0) = 0 \), the expression simplifies to \( \arctan\left(\frac{\sqrt{2}}{2}\right) \).

Step 5: Calculate Arctan Value

Evaluate \( \arctan\left(\frac{\sqrt{2}}{2}\right) \) using a calculator or table to find that it is approximately \( \frac{\pi}{8} \).

Key Concepts

Definite Integrals

When we talk about definite integrals, we're referring to the calculation of the area under a curve within a specified interval. In simple terms, it's like determining the total distance a car travels between two set points. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is noted by \( \int_{a}^{b} f(x) \, dx \). It results in a concrete number, as opposed to indefinite integrals which yield a family of functions.
Definite integrals have specific limits of integration, which are denoted as the numbers at the lower and upper bounds of the integral sign. These boundaries are crucial because they dictate over which interval we're finding the area under the curve.
To calculate a definite integral, follow these steps:

• Find the antiderivative of the function – this is akin to reversing differentiation.
• Apply the fundamental theorem of calculus: evaluate the antiderivative at the upper limit of integration and subtract its value at the lower limit.

In our exercise, we found the definite integral of \( \frac{\cos x}{1+\sin^2 x} \) from 0 to \( \frac{\pi}{4} \), arriving at the value after using trigonometric substitution.

Trigonometric Substitution

Trigonometric substitution is a nifty method for simplifying integrals involving roots and trigonometric expressions. By transforming variables in the integral with trigonometric functions, we make a new form that is easier to integrate. This method is incredibly useful when the integrand includes expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).
In the exercise we're looking at, we perform a substitution using the trigonometric identity: \( u = \sin x \) and, consequently, \( du = \cos x \, dx \). This transformation aids in simplifying the integral since it replaces the trigonometric part with a polynomial form.
Here's why you'd use this:

• Simplifying complex integrals: When expressions involve square roots or trigonometric identities, substitution can make them manageable.
• Transforming limits: Substitution changes the bounds of integration along with the variables, which needs careful computation.

This substitution turned our problem into an integral of \( \frac{1}{1+u^2} \), which points us directly to our next concept!

Arctan Function

The arctan function, also known as the inverse tangent function, is a key player in solving integrals of the form \( \int \frac{1}{1+u^2} \, du \). Recall that the standard outcome of such integrals is \( \arctan(u) + C \), where \( C \) is the constant of integration. The arctan function helps determine the angle whose tangent is a given number.
It's important to know:

• The range of the arctan function is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), so the output will always be within these bounds.
• Arctan provides an angle (in radians) that reflects the original tangent value. For example, \( \arctan(1) = \frac{\pi}{4} \).

In this exercise, after performing a substitution and solving the integral, we evaluated \( \arctan\left(\frac{\sqrt{2}}{2}\right) \). This final step is essential as it gives us the exact value of our definite integral within the specified limits.