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Find the integral $\int y^2e^{2y}dy$

Step-by-step Solution

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Final Answer

$\frac{1}{2}y^2e^{2y}-\frac{1}{2}ye^{2y}+\frac{1}{4}e^{2y}+C_0$
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Step-by-step Solution

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We can solve the integral $\int y^2e^{2y}dy$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$

$\begin{matrix}P(x)=y^2 \\ T(x)=e^{2y}\end{matrix}$

Learn how to solve integrals of exponential functions problems step by step online.

$\begin{matrix}P(x)=y^2 \\ T(x)=e^{2y}\end{matrix}$

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Learn how to solve integrals of exponential functions problems step by step online. Find the integral int(y^2e^(2y))dy. We can solve the integral \int y^2e^{2y}dy by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form \int P(x)T(x) dx. P(x) is typically a polynomial function and T(x) is a transcendent function such as \sin(x), \cos(x) and e^x. The first step is to choose functions P(x) and T(x). Derive P(x) until it becomes 0. Integrate T(x) as many times as we have had to derive P(x), so we must integrate e^{2y} a total of 3 times. With the derivatives and integrals of both functions we build the following table.

Final Answer

$\frac{1}{2}y^2e^{2y}-\frac{1}{2}ye^{2y}+\frac{1}{4}e^{2y}+C_0$

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1
2
3
4
5
6
7
8
9
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

How to improve your answer:

Main Topic: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

Used Formulas

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