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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:

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

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 $0$

$0$

Find the integral of $\sin\left(x\right)$ with respect to $x$

$\sin\left(x\right)$

Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$

$-\cos\left(x\right)$

The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function

$-\int\cos\left(x\right)dx$

Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$

$-\sin\left(x\right)$

The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function

$-\int\sin\left(x\right)dx$

Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$

$\cos\left(x\right)$
3

Integrate $T(x)$ as many times as we have had to derive $P(x)$, so we must integrate $\sin\left(x\right)$ a total of $3$ times

$\cos\left(x\right)$
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

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

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

$-x^2\cos\left(x\right)+2x\sin\left(x\right)+2\cos\left(x\right)+C_0$
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### beta Got another answer? Verify it!

Go!
1
2
3
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5
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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

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