Problem 9 Find the integral. $$ \int \... [FREE SOLUTION]

Chapter 4: Problem 9

Find the integral. $$ \int \frac{t}{\sqrt{1-t^{4}}} d t $$

Short Answer

Expert verified

The solution for the integral $\int \frac{t}{\sqrt{1-t^{4}}}\ dt$ is -$\frac{1}{2} \cos(2\arcsin(t^2)) + C$

Step by step solution

Identifying The Trigonometric Substitution

One can notice that $1 - t^4$ under the square root in the denominator is similar to $1 - x^2$ where $x$ is a trigonometric function. Here, the good substitution is $t^2 = \sin^2(z)$, thus $t = \sin(z)$ or $t = -\sin(z)$. We'll assume $t = \sin(z)$ for simplicity.

Calculate the Differentiation of t

To replace $dt$ in the integral, the derivative of $t$ with respect to $z$ is needed. The derivative $dt/dz$ of $\sin(z)$ is $\cos(z)$. Thus, $dt = \cos(z) dz$.

Transform the Integral

Substitute $t = \sin(z)$ and $dt = \cos(z) dz$ into the original integral. So the original integral becomes $\int \frac{sin(z)}{\sqrt{1-\sin^4(z)}} \cos(z) dz$. Then, simplify the integral to $\int \sin(z)cos(z) dz$.

Use the Trigonometric Identity

Recall that a well-known trigonometric identity is $\sin(2z) = 2 \sin(z)\cos(z)$. So the integral simplifies to $\frac{1}{2}\int \sin(2z) dz$.

Calculate the Integral

The integral of sine function is -cosine function. Therefore, the integral $\frac{1}{2}\int \sin(2z) dz$ is -$\frac{1}{2} \cdot \cos(2z) + C$, where $C$ is the constant of integration.

Back-Substitution

Rewrite $z$ in terms of $t$. The back-substitution $z = \arcsin(t^2)$ gives the final solution of the integral: $-\frac{1}{2} \cos(2\arcsin(t^2)) + C$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the integral. $$ \int \frac{t}{\sqrt{1-t^{4}}} d t $$

Short Answer

Step by step solution

Identifying The Trigonometric Substitution

Calculate the Differentiation of t

Transform the Integral

Use the Trigonometric Identity

Calculate the Integral

Back-Substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.