[amp_mcq option1=”ee-x” option2=”e-e-x” option3=”e-ex” option4=”e-x” correct=”option2″]<\/p>\n
The correct answer is $\\boxed{\\text{D}}$.<\/p>\n
The integral of $e^{-x} – e^{-x}$ is $e^{-x}$. This is because the derivative of $e^{-x}$ is $-e^{-x}$, so integrating $-e^{-x}$ gives $e^{-x}$ plus an arbitrary constant. Since we are integrating over the entire real line, the constant of integration must be zero.<\/p>\n
Here is a more detailed explanation:<\/p>\n
The integral of a function $f(x)$ is the area under the curve $y=f(x)$ between the $x$-axis and the line $x=a$, where $a$ is the upper limit of integration. In this case, $f(x)=e^{-x}-e^{-x}$ and $a=\\infty$.<\/p>\n
The area under the curve $y=e^{-x}-e^{-x}$ is shown in the following graph:<\/p>\n
[asy]
\nunitsize(1 cm);<\/p>\n
draw((0,0)–(10,0));
\ndraw((0,0)–(0,1));<\/p>\n
real ticklen=1;
\nreal tickspace=1;
\nreal axisarrowsize=0.14inch;
\nreal tickdown=-0.12inch;
\nreal tickdownlength=-0.12inch;
\nreal wholetickdown=-0.24inch;
\nreal wholetickdownlength=-0.24inch;
\nreal tickdownlengthshort=-0.06inch;
\nreal wholetickdownlengthshort=-0.12inch;
\nreal tickdownbase=0.12inch;
\nreal wholetickdownbase=0.24inch;
\nreal wholetickdownbaseshort=0.36inch;
\nreal t=0;
\nreal dt=1;
\nreal xleft=-2;
\nreal xright=12;
\nreal ybottom=-1;
\nreal ytop=1.2;<\/p>\n
label(“$x$”,(xright,0),E);
\nlabel(“$y$”,(0,ytop),N);<\/p>\n
real i;
\nfor (i=xleft; i<xright; i+=dt) {
\n draw((i,0)–(i,0.1));
\n}<\/p>\n
for (i=ybottom; i<ytop; i+=0.2) {
\n draw((0,i)–(0.1,i));
\n}<\/p>\n
draw((0,0)–(xright,0));
\ndraw((0,0)–(0,ytop));<\/p>\n
real f(real x) {
\n return exp(-x)-exp(-x);
\n}<\/p>\n
real g(real x) {
\n return exp(-x);
\n}<\/p>\n
draw(graph(f,xleft,xright),red);
\ndraw(graph(g,xleft,xright),blue);<\/p>\n
draw((xleft,f(xleft))–(xleft,0));
\ndraw((xright,f(xright))–(xright,0));
\ndraw((0,g(0))–(0,ytop));<\/p>\n
label(“$e^{-x}$”,(xright,f(xright)),E);
\nlabel(“$e^{-x}-e^{-x}$”,(xright,0),E);
\nlabel(“$e^{-x}$”,(0,g(0)),N);
\n[\/asy]<\/p>\n
The area under the curve is equal to the integral of $f(x)$ from $x=0$ to $x=\\infty$. This integral is equal to $e^{-x}$.<\/p>\n","protected":false},"excerpt":{"rendered":"
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