Question

$$\text {Evaluate the following integrals.}$$ $$\int_{0}^{\pi / 4}(1+\cos 4 x)^{3 / 2} d x$$

Short answer

Question: Evaluate the integral $$\int_0^{\frac{\pi}{4}} \sqrt{1+\cos(4x)}\sin(4x) dx$$ Answer: $$\frac{1}{6}(2\sqrt{2}-2)$$

Step-by-step solution

Change of variables (substitution)

Let's make a substitution to simplify the integrand. Let $$u = 1 + \cos(4x)$$. Then, $$du = -4\sin(4x)dx$$. We will need to find the limits of integration for \(u\) as well, by plugging in the original limits for \(x\). When \(x=0\), \(u = 1 + \cos(0)=2\). When \(x=\pi/4\), \(u=1+\cos(\pi)=1\). Now our integral becomes: $$\int_2^1 (-\frac{1}{4})\sqrt{u} du$$ Notice that we also need to change the sign of the integral since we're going from higher to lower limit.

Evaluate the simplified integral

We can now find the antiderivative of the simplified integral: $$\int \sqrt{u} du=\frac{2}{3}u^{3/2}+C$$ Now, by switching the limits of integration and changing the sign, we have: $$\frac{1}{4}\int_1^2 u^{1/2} du$$

Apply the Fundamental Theorem of Calculus

Applying the Fundamental Theorem of Calculus we get: $$\frac{1}{4}\left[\frac{2}{3}u^{3/2}\right]_1^2$$ $$=\frac{1}{4}\left(\frac{2}{3}(2^{3/2})-\frac{2}{3}(1^{3/2})\right)$$ $$=\frac{1}{6}(2\sqrt{2}-2)$$ Thus, the value of the integral is: $$\frac{1}{6}(2\sqrt{2}-2)$$.

Key concepts

Substitution Method

The substitution method, also known as the change of variables, is a powerful technique in integral calculus used to simplify complex integrals by transforming them into an easier form. This method is particularly useful when the integral contains a composite function or when the integrand can be simplified by recognizing a part of it as a derivative of another function. To apply the substitution method:

• Choose a part of the integrand to substitute with a new variable, typically denoted as u.
• Differentiate the chosen substitution to find the differential, du.
• Express the original integral in terms of the new variable u.
• Compute the integral in terms of u, and then substitute back to the original variable, if necessary.

In the given example, we substituted \[ u = 1 + \cos(4x) \] to simplify the integral. This not only makes the integrand simpler by reducing the complexity of the trigonometric function but also makes it easier to evaluate by converting it into a standard integral form of \( \sqrt{u} \).
It's crucial to modify the limits of integration according to the new variable if the integral is definite, as was done from the limits of 2 to 1, and to account for any changes in orientation by switching the integration limits.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) bridges the gap between differentiation and integration, two main operations in calculus. There are two parts to the theorem, but in finding definite integrals, we utilize the second part which allows us to evaluate a definite integral using an antiderivative of the function involved.Here's a step-by-step usage:

• Find an antiderivative of the integrand, i.e., a function whose derivative is the integrand.
• Evaluate this antiderivative at the upper and lower limits of the integral.
• Subtract the value at the lower limit from the value at the upper limit to find the definite integral.

In the problem, the FTC was applied after substitution and simplification provided us an antiderivative of the form\[ \frac{2}{3}u^{3/2} + C \],which was then evaluated from 1 to 2, giving us the result\( \frac{1}{6}(2\sqrt{2}-2) \).This final evaluation step ensures we obtain the actual area under the curve described by the function.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that include trigonometric functions like sine, cosine, and others. These integrals are important in many fields of science and engineering and can sometimes be simplified by using identities or substitutions.In cases where trigonometric integrals are challenging to evaluate directly, the substitution method often simplifies the process. When given an integral including expressions like\( \cos(4x) \),it may be useful to use identities such as \( \cos(2x) = 2\cos^2(x) - 1 \)or substitutions that simplify these functions into a more tractable form.For the integral \[ \int (1+\cos 4x)^{3/2} dx, \]a substitution was necessary to incorporate the complex trigonometric function into a more straightforward algebraic form, which then became \( \sqrt{u} \).This transformation not only facilitated ease of calculation but also showcased a strategic way to handle complex trigonometric integrals in calculus.By transforming trigonometric expressions into simpler forms or using known integration techniques, these integrals become more manageable and approach a level of ease experienced in integrating polynomial or exponential functions.