/*! 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 16 Find the indefinite integral by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 16

Find the indefinite integral by \(u\) -substitution. (Hint: Let \(u\) be the denominator of the integrand.) $$ \int \frac{1}{1+\sqrt{3 x}} d x $$

Short Answer

Expert verified
The indefinite integral of the function is \( \frac{2}{3} (1 + \sqrt{3x}) + C \).

Step by step solution

01

Identify and Set up the u-substitution

First, identify the denominator of the integral expression that would become more manageable with a substitution. In this exercise, the term inside the square root function (3x) is a suitable choice. If \( u = 1 + \sqrt{3x} \), we then take derivative of \( u \) to find \( du \). The derivative of \( u \) with respect to \( x \) is \( du = \frac{3}{2\sqrt{3x}} dx \).
02

Rearrange for dx

Rearrange the derivative equation to solve for \( dx \): \( dx = \frac{2}{3} u du \). This gives us a form to directly replace \( dx \) with \( du \) in our integral.
03

Substitute in the Integral

Now we substitute \( u \) for \( 1 + \sqrt{3x} \) and \( dx \) for \( \frac{2}{3} u du \) in our original integral. The new integral becomes \( \int \frac{1}{u} \cdot \frac{2}{3} u du = \frac{2}{3} \int du \).
04

Integrate and Back-substitute u

The integral \( \frac{2}{3} \int du \) simplifies to \( \frac{2}{3} u + C \), where \( C \) is the constant of integration. Finally, replace \( u \) with the original expression \( 1 + \sqrt{3x} \) to get \( \frac{2}{3} (1 + \sqrt{3x}) + C \).

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

Find the derivative of the function. \(y=\left(\operatorname{csch}^{-1} x\right)^{2}\)

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).

Verify the differentiation formula. \(\frac{d}{d x}\left[\operatorname{sech}^{-1} x\right]=\frac{-1}{x \sqrt{1-x^{2}}}\)

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \sqrt[4]{t} d t $$

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

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.