Question

Evaluate the following integrals. Include absolute values only when needed. $$\int_{0}^{\pi / 2} 4^{\sin x} \cos x d x$$

Short answer

Question: Evaluate the definite integral $$\int_{0}^{\pi / 2} 4^{\sin x} \cos x d x $$ using integration by substitution. Answer: $$\frac{3}{\ln(4)}$$

Step-by-step solution

Identify the substitution

We will make a substitution \(u = \sin x\). This will help simplify the expression inside the integral.

Calculate the derivative of the substitution

Calculate the derivative of the substitution with respect to x. In other words, find \(\frac{du}{dx}\) for \(u = \sin x\), which is \(\frac{du}{dx} = \cos x\).

Convert the integral to terms of u

We can rewrite the provided integral $$\int_{0}^{\pi / 2} 4^{\sin x} \cos x d x $$ as $$\int_{u(\pi/2)}^{u(0)} 4^{u} d u $$. To find the limits of integration, we can substitute \(x = 0\) for the lower limit and \(x = \pi / 2\) for the upper limit: $$u(0) = \sin(0) = 0$$ $$u(\pi/2) = \sin(\pi/2) = 1$$ Now, our integral becomes: $$\int_{0}^{1} 4^{u} d u$$

Evaluate the integral using substitution

To evaluate the integral, we can make use of the following formula: $$\int e^{au} d u = \frac{1}{a} e^{au} + C$$ Replace \(e\) with 4 and \(a\) with \(\ln(4)\): $$\int 4^u d u = \int e^{\ln(4)u} d u = \frac{1}{\ln(4)} e^{\ln(4) u} + C$$ Now, we can compute the definite integral: $$\int_{0}^{1} 4^{u} d u = \left[\frac{1}{\ln(4)} e^{\ln(4) u} \right]_{0}^{1}$$ which is equal to $$\frac{1}{\ln(4)} \left( e^{\ln(4)} - e^{0} \right) = \frac{1}{\ln(4)} (4 - 1) = \frac{3}{\ln(4)}$$

Convert back to the original variable

Since our integral is now a definite integral with fixed limits, there is no need to convert back to the original variable. The solution is: $$\int_{0}^{\pi / 2} 4^{\sin x} \cos x d x = \frac{3}{\ln(4)}$$

Key concepts

Definite Integral

A definite integral represents the area under a curve within certain limits on a graph. Unlike an indefinite integral, which gives a general form of an antiderivative, a definite integral evaluates the total area between the curve of a function, the x-axis, and the vertical lines at the limits of integration.
For example, in the given integral from the exercise, we have limits from 0 to \(\pi/2\):

• Lower limit: \(x = 0\)
• Upper limit: \(x = \pi/2\)

The main goal is to compute the "net area" under the curve of the function \(4^{\sin x} \cos x\) within those limits.
After solving, the result is a specific numerical value, which in this case, reflects that area under the curve from point 0 to \(\pi/2\) on the x-axis.

Substitution Method

The substitution method is a powerful technique used in solving integrals, especially when dealing with compositions of functions that are otherwise challenging to handle directly. This method involves substituting parts of the original integral with a new variable to simplify the integration process.
Here's a quick breakdown of the process:

• Choose a Substitution: Identify a part of the integral that can simplify the problem if substituted. In our exercise, \(u = \sin x\) is chosen.
• Calculate Derivative: Compute \(\frac{du}{dx}\) for your \( u \), resulting in \(\frac{du}{dx} = \cos x\), hence \(du = \cos x \, dx\).
• Change Limits: Convert the limits from \(x\) to \(u\). This step ensures all parts of the integral are in terms of \(u\).
• Evaluate the New Integral: Solve the integral in terms of \(u\), and convert back if needed. For definite integrals, once evaluated, conversion back to the original variable is not necessary.

In this exercise, using substitution made it easier to transform and compute the integral \(\int_{0}^{\pi / 2} 4^{\sin x} \cos x \, dx\) to \(\int_{0}^{1} 4^u \, du\), simplifying the solving process greatly.

Trigonometric Substitution

Trigonometric substitution is a special case of the substitution method where trigonometric identities are employed to simplify integrals, especially those involving radicals or products of trigonometric functions.
It works best in situations where the integral involves:

• Sine and cosine functions, as in \(\sin x\) and \(\cos x\), which inherently offer substitution potential due to their derivability relationship.
• Expressions that can benefit from the simplification using trigonometric identities.

In our example, choosing \(u = \sin x\) capitalizes on the trigonometric relationship \(\frac{d}{dx}(\sin x) = \cos x\). This allows seamless substitution to clear the \(\cos x \, dx\) from the integral, streamlining the process of evaluation.
By reducing the original integral into one involving a simpler exponential function \(4^u\), trigonometric substitution proves to be a smart strategy for tackling otherwise daunting problems in integration.