/*! 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 10 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

Find the integral. $$ \int \frac{t}{t^{4}+16} d t $$

Short Answer

Expert verified
The integral \(\int \frac{t}{t^{4}+16} dt = \frac{1}{4}\arctan(\frac{t^{2}}{4})\)

Step by step solution

01

Choose a substitution

Choose \(u = t^{2}\). Hence, \(du = 2t dt\).
02

Rewrite the integral in terms of u

Divide du by 2 and substitute in the integral. This turns the integral into \(\frac{1}{2}\int \frac{1}{u^{2}+16} du\).
03

Simplify the integral

This is a standard integral which can be solved as \(\arctan(\frac{x}{a}) = \frac{1}{a}\arctan(\frac{x}{a})\). Apply this to get \(\frac{1}{4}\arctan(\frac{u}{4})\).
04

Substitute back

Substitute \(u = t^{2}\) back into the integral to get the final result. The result is \(\frac{1}{4}\arctan(\frac{t^{2}}{4})\).

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!

Key Concepts

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

Substitution Method
The Substitution Method is an essential technique in integral calculus that helps simplify complex integrals. Think of it like changing variables to make the problem easier to solve. In this method, you choose a part of the integral to substitute with a new variable.
  • Start by identifying a part of your integral that can be replaced. This is usually the entire expression in a complex term.
  • Next, you express the differential of the new variable (like \( du \)) in terms of the original variable (like \( dt \)).
In our example, we set \( u = t^{2} \) which transforms the term \( t \cdot dt \) to \( \frac{1}{2} du \). This substitution simplifies the original integral, making it much more manageable to solve.
Remember, the goal of substitution is to transform the integral into a standard form that you know how to integrate easily.
Trigonometric Integration
Trigonometric Integration involves using trigonometric identities and functions to solve integrals that include trigonometric terms. In our exercise, once the substitution is made, we transform the integrand into a standard form that involves trigonometric functions.
  • Trigonometric identities, like \( \arctan \), are often used to solve these integrals.
  • Our integral turns into \( \frac{1}{u^{2} + 16} \), a familiar trigonometric form that relates to arctan function.
This specific form can be integrated using the formula \( \int \frac{1}{x^{2} + a^{2}} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C \), where \( a \) is a constant. This shows how knowledge of trigonometric integrals can aid in solving seemingly complex problems with ease.
Definite and Indefinite Integrals
In calculus, integrals come in two types: definite and indefinite. Each kind offers different information and solutions.
  • Indefinite Integrals: These represent a family of functions and include a constant \(C\). They indicate anti-derivatives or functions who derivative is the integrand.
  • Definite Integrals: These provide a numeric value representing the area under a curve between two points on a graph. They have upper and lower limits.

In the exercise provided, we are focusing on finding the indefinite integral, as there are no limits of integration specified. This means that we find a function whose derivative is the integrand, plus the constant of integration \(C\). In practice, indefinite integrals are crucial as they provide the general solution before any boundary or conditions are applied.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Prove that \(\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1

Verify the differentiation formula. $$ \frac{d}{d x}[\cosh x]=\sinh x $$

Discuss several ways in which the hyperbolic functions are similar to the trigonometric functions.

Find the limit. $$ \lim _{x \rightarrow \infty} \tanh x $$

Evaluate the integral. $$ \int_{0}^{4} \frac{1}{25-x^{2}} d x $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.