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# Find the integral $\int x^2\sin\left(2x\right)dx$

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##  Final answer to the problem

$-\frac{1}{2}x^2\cos\left(2x\right)+\frac{1}{2}x\sin\left(2x\right)+\frac{1}{4}\cos\left(2x\right)+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Integrate by trigonometric substitution
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
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1

We can solve the integral $\int x^2\sin\left(2x\right)dx$ 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 $T(x)$

$\begin{matrix}P(x)=x^2 \\ T(x)=\sin\left(2x\right)\end{matrix}$

Learn how to solve integral calculus problems step by step online.

$\begin{matrix}P(x)=x^2 \\ T(x)=\sin\left(2x\right)\end{matrix}$

Learn how to solve integral calculus problems step by step online. Find the integral int(x^2sin(2x))dx. We can solve the integral \int x^2\sin\left(2x\right)dx 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 \sin\left(2x\right) a total of 3 times. With the derivatives and integrals of both functions we build the following table.

##  Final answer to the problem

$-\frac{1}{2}x^2\cos\left(2x\right)+\frac{1}{2}x\sin\left(2x\right)+\frac{1}{4}\cos\left(2x\right)+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

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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

###  Main Topic: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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