/*! 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 22 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

Find the indefinite integral. $$ \int \frac{\csc ^{2} t}{\cot t} d t $$

Short Answer

Expert verified
The indefinite integral is \( - 2 \cot t + C \)

Step by step solution

01

Rewrite the integral in terms of sine and cosine

Rewrite the integral as \( \int \frac{1}{\sin^2t \cdot \cos t} dt \). We can rewrite \( \csc^2t \) as \( \frac{1}{\sin^2t} \) and \( \cot t \) as \( \cos t \), thus the integral becomes \( \int \frac{1}{\sin^2t \cdot \cos t} dt \).
02

Simplification

Split the integral as \( \int \frac{1}{\sin^2t \cdot \cos t} dt \) changing to \( \int \frac{\cos t}{\sin^2t} dt - \int \frac{1}{\sin^2t} dt \).
03

Solve the integrals

The first integral is the derivative of cotangent, thus, \( - \cot t \). The second integral is equal to \( \csc^2 t \) which is \( -\cot t + C \). Therefore, the solution is sum of both namely, \( - 2 \cot t + C \).

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

Most popular questions from this chapter

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} \sec ^{3} t d t $$

Consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(\int_{a}^{b}\left|x^{\prime}(t)\right| d t\) is the distance the particle travels in the interval of time. A particle moves along the \(x\) -axis with velocity \(v(t)=1 / \sqrt{t}\) \(t > 0\). At time \(t=1,\) its position is \(x=4\). Find the total distance traveled by the particle on the interval \(1 \leq t \leq 4\).

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

Find any relative extrema of the function. Use a graphing utility to confirm your result. \(g(x)=x \operatorname{sech} x\)

Show that \(\arctan (\sinh x)=\arcsin (\tanh x)\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.