4.2.1 AP calc exam another int with e

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In summary, the integral $\displaystyle\int\dfrac{e^{2x}}{1+e^x} \, dx$ can be solved using the substitution $u=1+e^x$, resulting in the answer $e^x - \ln(1+e^x) + C$, with the choices $a.\; \tan^{-1}e^x+C$, $b.\; 1+e^x-\ln(1+e^1)+C$, $c.\; x-x+\ln |1+e^x|+C$, $d.\; e^x+\frac{1}{(e^x+1)^2}+C$, and $e.\
  • #1
karush
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Evaluate $\displaystyle\int\dfrac{e^{2x}}{1+e^x} \, dx=$

$a.\quad \tan^{-1}e^x+C$
$b.\quad 1+e^x-\ln(1+e^1)+C$
$c.\quad x-x+\ln |1+e^x|+C$
$d.\quad e^x+\frac{1}{(e^x+1)^2}+C$
$e.\quad {none}$

ok I was going to use $u=1+e^x\quad du=e^x dx$ but maybe not best

btw I tried to use array on the choices but its was all underlined in preview
 
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  • #2
l$u=e^x \implies du = e^x \, dx$

$\displaystyle \int \dfrac{u}{1+u} \, du$

$\displaystyle \int \dfrac{u+1-1}{1+u} \, du$

$\displaystyle \int 1 - \dfrac{1}{1+u} \, du$

$u - \ln(1+u) + C$

$e^x - \ln(1+e^x) + C$

btw ... AP exams never use “none” as a choice
 
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  • #3
yeah I know but typically they have 5 not 4 choices Ill invent some bogus answer

that is a very rare way to solve it not sure if I even saw that unless you are completing the square
 
  • #4
that is a very rare way to solve it not sure if I even saw that unless you are completing the square

?
karush said:
Evaluate $\displaystyle\int\dfrac{e^{2x}}{1+e^x} \, dx=$

ok I was going to use $u=1+e^x\quad du=e^x dx$ but maybe not best

that works, too ...

$u = 1+e^x \implies du = e^{x} \, dx \text{ and } e^x = u-1$

\(\displaystyle \int \dfrac{u-1}{u} \, du\)

\(\displaystyle \int 1 - \dfrac{1}{u} \, du\)

$u - \ln(u) + C$

$1+e^x - \ln(1+e^x) + C$
 
  • #5
skeeter said:
l$u=e^x \implies du = e^x \, dx$

$\displaystyle \int \dfrac{u}{1+u} \, du$
The given problem had $e^{2x}= (e^x)^2$ in the denominator so the integral is
$\int \dfrac{u^2}{1+ u}{du}= \int 1- \dfrac{1}{1+ u} du$$= u- ln|1+ u|+ C= e^x- ln|1+ e^x|+ C$

$\displaystyle \int \dfrac{u+1-1}{1+u} \, du$

$\displaystyle \int 1 - \dfrac{1}{1+u} \, du$

$u - \ln(1+u) + C$

$e^x - \ln(1+e^x) + C$

btw ... AP exams never use “none” as a choice
 
  • #6
Country Boy said:
The given problem had $e^{2x}= (e^x)^2$ in the denominator so the integral is
$\int \dfrac{u^2}{1+ u}{du}= \int 1- \dfrac{1}{1+ u} du$$= u- ln|1+ u|+ C= e^x- ln|1+ e^x|+ C$

huh?

karush said:
Evaluate $\displaystyle \int\dfrac{e^{2x}}{{\color{red}1+e^x}} \, dx=$

$a.\quad \tan^{-1}e^x+C$
$b.\quad 1+e^x-\ln(1+e^1)+C$
$c.\quad x-x+\ln |1+e^x|+C$
$d.\quad e^x+\frac{1}{(e^x+1)^2}+C$
$e.\quad {none}$

ok I was going to use $u=1+e^x\quad du=e^x dx$ but maybe not best

btw I tried to use array on the choices but its was all underlined in preview
 

Related to 4.2.1 AP calc exam another int with e

1. What is the format of the 4.2.1 AP Calc exam?

The 4.2.1 AP Calc exam consists of a multiple-choice section and a free-response section. The multiple-choice section contains 45 questions and is 1 hour and 45 minutes long. The free-response section contains 6 questions and is 1 hour and 30 minutes long.

2. What topics are covered on the 4.2.1 AP Calc exam?

The 4.2.1 AP Calc exam covers topics such as limits, derivatives, integrals, and applications of derivatives and integrals. It also includes topics such as differential equations, sequences and series, and vector functions.

3. How is the 4.2.1 AP Calc exam scored?

The multiple-choice section of the 4.2.1 AP Calc exam is scored on a scale of 1-5, with 5 being the highest score. The free-response section is also scored on a scale of 1-5. The two scores are then combined to give a final score out of 5.

4. Is the use of a calculator allowed on the 4.2.1 AP Calc exam?

Yes, a graphing calculator is allowed on the 4.2.1 AP Calc exam. However, there are certain restrictions on the types of calculators that can be used. The College Board provides a list of approved calculators that can be used on the exam.

5. How should I prepare for the 4.2.1 AP Calc exam?

To prepare for the 4.2.1 AP Calc exam, it is important to review all of the topics covered in the course and to practice solving problems similar to those that will be on the exam. It is also helpful to take practice exams and to familiarize yourself with the format and timing of the exam.

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