Challenging Integrals: Can You Solve These Three?

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In summary, the conversation discusses three different integrals and strategies for solving them. The first two integrals involve using substitutions to simplify the integrand, while the third one requires using trigonometric substitution. The conversation also mentions that no integration by parts or partial fractions are allowed for these integrals.
  • #1
clairez93
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Homework Statement



1. [tex]\int\frac{2}{e^{-x} + 1}[/tex]

[tex]\int\frac{2t - 1}{t^{2} + 4}[/tex]

[tex]\int\frac{4}{4x^{2} + 4x + 65}[/tex]

Homework Equations





The Attempt at a Solution



1. I'm not sure what to do. A u-substitution would give me e^(-x) dx, but I don't see how I could get an e^(-x) dx anywhere in the integrand.

2. Again I have no idea. I'm thinking maybe it will involve arc tan, except the numerator isn't 1, so what would I do with the 2t-1 numerator?

3. I tried completing the square, to get something that resembled an arc tan, but that didn't work too well. I'm not sure what else to do; a u-substitution wouldn't work I don't think that would just get 8x + 4.

No integration by parts or partial fractions allowed yet. Any help appreciated.
 
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  • #2
For the first one, note that [tex]\int\frac{2}{e^{-x} + 1}\,dx=\int \frac{2}{1+\frac{1}{e^x}}}\,dx[/tex].

For the second one [tex]\int\frac{2t - 1}{t^{2} + 4}\,dt = \int\frac{2t}{t^{2} + 4}\,dt - \int\frac{1}{t^{2} + 4}\,dt[/tex]

For the third one, you almost have it. Try doing 2x+1 = 8tanθ
 
  • #3
Thanks for the other two hints, I don't understand what you mean by 2x+1 = tan theta?
 
  • #4
Are you familiar with trigonometric substitution?
[tex]\int\frac{4}{4x^{2} + 4x + 65}\,dx=\int\frac{4}{(2x+1)^{2}+8^{2}}\,dx=\frac{1}{16}\int\frac{1}{\tan^{2}(\theta)+1}\,dx[/tex]

because
[tex]8\tan(\theta)=2x+1[/tex], [tex]\theta=\arctan(\frac{2x+1}{8})[/tex] and [tex]2dx=8\sec^{2}(\theta)[/tex].
 
  • #5
Oooh nevermind I got it
 

What are three integrals to evaluate?

Three commonly used integrals to evaluate are the definite integral, indefinite integral, and line integral.

What is a definite integral?

A definite integral is a type of integral that has specific limits of integration, or values for which the integral will be evaluated. It represents the area under a curve between two points on a graph.

What is an indefinite integral?

An indefinite integral is a type of integral that does not have specific limits of integration. It represents a family of functions that differ by a constant value, and is often used to find the original function from a derivative.

What is a line integral?

A line integral is an integral that is calculated along a curve or line in a multi-dimensional space. It is often used in physics and engineering to calculate work done or measure flow rates.

What are some applications of integrals?

Integrals have many practical applications in fields such as physics, engineering, economics, and statistics. Some common applications include calculating areas and volumes, finding distances and displacements, and solving differential equations.

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