/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 \(\int_{\pi}^{5 \pi / 4} \frac{\... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 19

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

Short Answer

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

Step by step solution

01

Identify the substitution

To simplify the integral, use the trigonometric identity substitution. Let鈥檚 set \(u = 2x\) Then, \(du = 2dx\) or \(dx = \frac{du}{2}\).
02

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 \(x = \frac{5\pi}{4}\), \(u = \frac{5\pi}{2}\).
03

Transform the integral

The integral becomes: \[ \int_{2\pi}^{\frac{5\pi}{2}} \frac{\sin u}{\cos^{4}\left(\frac{u}{2}\right) + \sin^{4}\left(\frac{u}{2}\right)} \cdot \frac{du}{2} \].
04

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.
05

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 \(\frac{\pi}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Two bodies of different materials and same surface area, coated by similar material, are heated to same temperature. They are allowed to cool down in same surroundings. If \(\frac{d Q}{d t}\) represents rate of loss of heat and \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) represents rate of fall of temperature, then for both : (1) \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is same but \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) is different (2) \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is different but \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) is same (3) \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) as well as \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is same (4) \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) as well as \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) is different

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If \(x=5\) and \(y=-2\) then \(x-2 y=9\). The contrapositive of this statement is (1) If \(x-2 y=9\) then \(x=5\) and \(y=-2\) (2) If \(x-2 y \neq 9\) then \(x \neq 5\) and \(y \neq-2\) (3) If \(x-2 y \neq 9\) then \(x \neq 5\) or \(y \neq-2\) (4) If \(x-2 y \neq 9\) then either \(x \neq 5\) or \(y=-2\)

Solution of the differential equation \(\frac{d y}{d x}+\frac{1+y^{2}}{\sqrt{1-x^{2}}}=0\) is (1) \(\tan ^{-1} y+\sin ^{-1} x=c\) (2) \(\tan ^{-1} x+\sin ^{-1} y=c\) (3) \(\tan ^{-1} y \cdot \sin ^{-1} x=c\) (4) \(\tan ^{-1} \mathrm{y}-\sin ^{-1} x=c\)
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