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

## Step-by-step Solution

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###  Videos

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

Problem to solve:

$\int x^2\sin\left(x\right)dx$

Specify the solving method

1

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

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

Find the derivative of $x^2$ with respect to $x$

$x^2$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$

The derivative of the linear function times a constant, is equal to the constant

$2$

The derivative of the constant function ($2$) is equal to zero

0
2

Derive $P(x)$ until it becomes $0$

$0$
3 Try to guess Step 3. Or become premium for the price of a latte.
4

With the derivatives and integrals of both functions we build the following table

$\begin{matrix}\mathrm{Derivatives} & \mathrm{Sign} & \mathrm{Integrals} \\ & & \sin\left(x\right) \\ x^2 & + & -\cos\left(x\right) \\ 2x & - & -\sin\left(x\right) \\ 2 & + & \cos\left(x\right) \\ 0 & & \end{matrix}$
5

Then the solution is the sum of the products of the derivatives and the integrals according to the previous table. The first term consists of the product of the polynomial function by the first integral. The second term is the product of the first derivative by the second integral, and so on.

$-x^2\cos\left(x\right)+2x\sin\left(x\right)+2\cos\left(x\right)$
6 Try to guess Step 6. Or become premium for the price of a latte.

$-x^2\cos\left(x\right)+2x\sin\left(x\right)+2\cos\left(x\right)+C_0$

##  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

Solve int(x^2sin(x))dx using basic integralsSolve int(x^2sin(x))dx using u-substitutionSolve int(x^2sin(x))dx using integration by parts

SnapXam A2

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7
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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

### Main topic:

Integral Calculus

~ 0.02 s

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