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Find the integral $\int\frac{\sin\left(x\right)^2}{e^{-x}}dx$

Step-by-step Solution

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Final Answer

$-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}x-\frac{1}{8}\cos\left(2x\right)-\frac{1}{4}x^2+\frac{1}{4}x\left(2x-\sin\left(2x\right)\right)+\frac{1}{12}x^{3}-\frac{1}{8}x\left(2x^2+\cos\left(2x\right)\right)+\frac{1}{8}x^{2}\left(2x-\sin\left(2x\right)\right)-\frac{1}{256}u^{4}+\frac{1}{64}u^{3}\sin\left(u\right)+\frac{1}{192}\left(2x\right)^{3}\left(2x-\sin\left(2x\right)\right)+C_0$
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Step-by-step Solution

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1

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int e^{-\left(-1\right)x}\sin\left(x\right)^2dx$

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$\int e^{-\left(-1\right)x}\sin\left(x\right)^2dx$

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Learn how to solve problems step by step online. Find the integral int((sin(x)^2)/(e^(-x)))dx. Rewrite the exponent using the power rule \frac{a^m}{a^n}=a^{m-n}, where in this case m=0. Multiply -1 times -1. Use the Taylor series for rewrite the function e^x as an approximation: \displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n, with a=0. Here we will use only the first four terms of the serie to approximate the function. Any expression divided by one (1) is equal to that same expression.

Final Answer

$-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}x-\frac{1}{8}\cos\left(2x\right)-\frac{1}{4}x^2+\frac{1}{4}x\left(2x-\sin\left(2x\right)\right)+\frac{1}{12}x^{3}-\frac{1}{8}x\left(2x^2+\cos\left(2x\right)\right)+\frac{1}{8}x^{2}\left(2x-\sin\left(2x\right)\right)-\frac{1}{256}u^{4}+\frac{1}{64}u^{3}\sin\left(u\right)+\frac{1}{192}\left(2x\right)^{3}\left(2x-\sin\left(2x\right)\right)+C_0$

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Function Plot

Plotting: $-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}x-\frac{1}{8}\cos\left(2x\right)-\frac{1}{4}x^2+\frac{1}{4}x\left(2x-\sin\left(2x\right)\right)+\frac{1}{12}x^{3}-\frac{1}{8}x\left(2x^2+\cos\left(2x\right)\right)+\frac{1}{8}x^{2}\left(2x-\sin\left(2x\right)\right)-\frac{1}{256}u^{4}+\frac{1}{64}u^{3}\sin\left(u\right)+\frac{1}{192}\left(2x\right)^{3}\left(2x-\sin\left(2x\right)\right)+C_0$

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7
8
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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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