Problem 50 - Calculus Early Transcendentals

Question

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2 x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2 x} d x$$ Answer: The value of the integral is 0.

Step 1: Use integration by substitution

Let's make the substitution: $$u = \sin 2x$$ Now, we need to find the derivative of u with respect to x: $$\frac{d u}{d x} = \frac{d(\sin 2x)}{d x}= 2\cos{2x}$$ So, the differential du is: $$du = 2\cos{2x} dx$$ Now we can rewrite the integral in terms of u and its differential du: $$\int 3 \sqrt{1+u} \frac{1}{2}du$$

Step 2: Simplify the integral

Now, we will simplify the integral and perform the integration: $$\frac{3}{2} \int (1+u)^{1/2} du$$ To evaluate the integral, we will use the power rule for integration, which states that for any function of the form f(x)=x^n, its integral is F(x)=((x^(n+1))/(n+1))+C, where C is the constant of integration. In our case, we have: $$\frac{3}{2} \int (1+u)^{1/2} du = \frac{3}{2} \left[\frac{(1+u)^{3/2}}{3/2}\right] + C$$

Step 3: Replace the substitution variable

Now, we need to replace the variable u with the original trigonometric function: $$\frac{3}{2}\left[\frac{(1+\sin{2x})^{3/2}}{3/2}\right] + C$$ Simplify the expression: $$ (1+\sin{2x})^{3/2} + C$$ Now we will find the definite integral. For this, we need to evaluate the function between the given limits, 0 and π/4.

Step 4: Evaluate the definite integral

To evaluate the integral, we need to find the difference between the values of the function at the upper and lower bound: $$\eval {(1+\sin{2x})^{3/2}}_{0}^{\frac{\pi}{4}}$$ Now, substitute the limits: $$\left[(1+\sin{(\pi)})^{3/2} - (1+\sin{0})^{3/2}\right]$$ Simplify and solve: $$\left[(1+0)^{3/2} - (1+0)^{3/2}\right] = 0$$ In conclusion, the value of the integral: $$\int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2 x} d x = 0$$

Key Concepts

Integration by Substitution

Integration by substitution, often referred to as u-substitution, is a technique to simplify certain integrals by making a substitution that transforms a complex integral into a simpler one. This method is particularly useful when dealing with integrals of functions composed with other functions.

For instance, consider the integral from the exercise: \[ \int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2x} \, dx \]. Here, by setting \( u = \sin 2x \), we translate the problem into the u-domain, which often makes integration more straightforward. The derivative of u with respect to x is found, \( \frac{du}{dx} = 2\cos{2x} \), and the original integral is re-expressed in terms of u as \( \frac{3}{2} \int \sqrt{1+u} \, du \).

Why Use Substitution?

Substitution simplifies integration by reducing complex expressions to more basic forms. In some cases, it can convert an integral into a standard form, where a known formula applies, much like in our exercise where the power rule is applicable post-substitution. This method is particularly useful when direct integration is not feasible or when the integrand involves a chain of functions.

Trigonometric Integration

Trigonometric integration involves techniques for integrating functions that contain trigonometric expressions. Integral problems that feature sine, cosine, or other trigonometric functions can sometimes pose a challenge due to their oscillatory nature.

In the provided exercise, we have the function \( \sqrt{1 + \sin 2x} \) inside the integral. Trigonometric integration techniques can be employed here, such as using trigonometric identities to simplify the expression or using substitution to integrate terms involving trigonometric functions, which was demonstrated in the solution. After the initial substitution, we no longer have to deal with the trigonometric function directly until we back-substitute.

The Role of Limits in Trigonometric Integration

Trigonometric integration with definite integrals requires attention to the limits of integration. When substituting, the limits should also be converted to match the new variable, unless we substitute back to the original variable before evaluating, as seen in our solution where we reverted to x to calculate the definite integral.

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus integral rules. It states that for any real number n, except -1, the integral of \( x^n \) with respect to x is \( \frac{x^{n+1}}{n+1} \), plus a constant of integration C.

In our exercise, after using substitution, we applied the power rule to integrate \( (1+u)^{1/2} \). The integral became \( \frac{3}{2} \int (1+u)^{1/2} du \), which follows the power rule pattern and integrates to \( \frac{3}{2} [\frac{(1+u)^{(1/2)+1}}{(1/2)+1}] + C \), or simplified, \( (1+u)^{3/2} + C \).

Applicability of the Power Rule

The power rule is widely used because it applies to any power of x, and many functions can be manipulated into a form where the power rule is applicable. This rule is one of the first techniques students learn for integrating polynomials and plays a crucial role in solving a wide range of problems in calculus.