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Find the integral $\int x\sin\left(x\right)^3\cos\left(x\right)dx$

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Solution

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Final answer to the problem

$-x\cos\left(x\right)+\frac{x\cos\left(x\right)^{3}}{3}+\frac{7}{9}\sin\left(x\right)+\frac{-\cos\left(x\right)^{2}\sin\left(x\right)}{9}+C_0$
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Step-by-step Solution

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  • 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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We can solve the integral $\int x\sin\left(x\right)^3\cos\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

Learn how to solve definition of derivative problems step by step online.

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

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Unlock the first 3 steps of this solution

Learn how to solve definition of derivative problems step by step online. Find the integral int(sin(x)^3xcos(x))dx. We can solve the integral \int x\sin\left(x\right)^3\cos\left(x\right)dx by applying integration by parts method to calculate the integral of the product of two functions, using the following formula. First, identify or choose u and calculate it's derivative, du. Now, identify dv and calculate v. Solve the integral to find v.

Final answer to the problem

$-x\cos\left(x\right)+\frac{x\cos\left(x\right)^{3}}{3}+\frac{7}{9}\sin\left(x\right)+\frac{-\cos\left(x\right)^{2}\sin\left(x\right)}{9}+C_0$

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Function Plot

Plotting: $-x\cos\left(x\right)+\frac{x\cos\left(x\right)^{3}}{3}+\frac{7}{9}\sin\left(x\right)+\frac{-\cos\left(x\right)^{2}\sin\left(x\right)}{9}+C_0$

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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
D□x
∫
∫◻
|◻|
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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Similar Problems Solved

$derivdef\left(x<y<0\right)$

Main Topic: Definition of Derivative

Resolution of derivatives using the definition of the derivative, which is the limit of difference quotients of real numbers.

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