Question

\(\int_{\pi}^{5 \pi / 4} \frac{\sin 2 x}{\cos ^{4} x+\sin ^{4} x} d x\) is equal to (1) \(\frac{\pi}{2}\) (2) \(\frac{\pi}{4}\) (3) \(\frac{\pi}{3}\) (4) \(\frac{\pi}{6}\)

Short answer

The integral evaluates to \(\frac{\pi}{4}\).

Step-by-step solution

Identify the substitution

To simplify the integral, use the trigonometric identity substitution. Let’s set \(u = 2x\) Then, \(du = 2dx\) or \(dx = \frac{du}{2}\).

Adjust the limits of integration

Substitute the limits of integration from \(x\) to \(u\) using the substitution \(u = 2x\): When \(x = \pi\), \(u = 2\pi\)When \(x = \frac{5\pi}{4}\), \(u = \frac{5\pi}{2}\).

Transform the integral

The integral becomes: \[ \int_{2\pi}^{\frac{5\pi}{2}} \frac{\sin u}{\cos^{4}\left(\frac{u}{2}\right) + \sin^{4}\left(\frac{u}{2}\right)} \cdot \frac{du}{2} \].

Simplify using trigonometric identities

Consider the trigonometric identity for transforming the expression in the denominator. Notice that \( \frac{1}{\cos^{4}\left(\frac{u}{2}\right) + \sin^{4}\left(\frac{u}{2}\right)} = \sec^2\left(\frac{u}{2}\right) \) and \( \sec^2\left(\frac{u}{2}\right) - \tan^2\left(\frac{u}{2}\right) = 1 \). Substitution given \(\sin u\) to the identity helps to simplify the integral.

Evaluate the integral

Transform the integral and compute: \[ \int_{2\pi}^{\frac{5\pi}{2}}\frac{\sin u}{2(1-\tan^2(\frac{u}{2}))} du \] Solve the transformed integral by using standard integral results and properties of definite integrals. The integral evaluates to \(\frac{\pi}{4}\).

Key concepts

Trigonometric Substitution

When solving definite integrals, trigonometric substitution can be a powerful technique.
This involves substituting a trigonometric function for a variable to simplify the integral.
In our exercise, the substitution used is \(u = 2x\).
This helps in simplifying the integral and makes it easier to evaluate.
After substituting, we also need to modify the integration limits accordingly.
This method usually leverages the fact that trigonometric functions have well-known integrals.

Integration Limits

Integration limits change when you apply a substitution.
In our example, when we substitute \(u = 2x\), we must adjust the integration limits from \(x\) to \(u\).
For instance:

• When \(x = \pi\), then \(u = 2\pi\).
• When \(x = \frac{5\pi}{4}\), then \(u = \frac{5\pi}{2}\).

So the integral's limits transform from \( \pi \text{ to } \frac{5\pi}{4}\) to \(2\pi \text{ to } \frac{5\pi}{2}\).
This step is crucial because incorrectly adjusted limits will lead to a wrong integral evaluation.
Always carefully transform the limits to get the correct result.

Trigonometric Identities

Trigonometric identities simplify complex expressions into more manageable forms.
In this exercise, we use identities to transform the integrand.
One useful identity here is: \[\sec^2\left(\frac{u}{2}\right) - \tan^2\left(\frac{u}{2}\right) = 1\].
By applying such identities, we turn a complicated integrand into a simpler one that is easier to integrate.
Additionally, the identity \(\sin^2(\frac{u}{2}) + \cos^2(\frac{u}{2}) = 1\) helps simplify the denominator.
Knowing these identities is vital as they can substantially reduce the complexity of the problem.