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

Step-by-step Solution

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Solution

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Final Answer

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

Problem to solve:

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

Specify the solving method

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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

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

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

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

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Unlock the first 2 steps of this solution.

Learn how to solve integral calculus 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 u and calculate du. Now, identify dv and calculate v. Solve the integral.

Final Answer

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

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 integral of sinx^3xdx using basic integralsSolve integral of sinx^3xdx using u-substitutionSolve integral of sinx^3xdx using integration by parts

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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
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

How to improve your answer:

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Main topic:

Integral Calculus

Used formulas:

9. See formulas

Related topics:

Integral CalculusCalculusIntegration by PartsIntegration Techniques

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