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Find the integral $\int\frac{x^4+20x^3+150x^2}{12}\sin\left(3x\right)dx$

Step-by-step Solution

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Final Answer

$-\frac{1}{36}x^4\cos\left(3x\right)-\frac{5}{9}x^3\cos\left(3x\right)-\frac{223}{54}x^{2}\cos\left(3x\right)+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{5}{9}x^{2}\sin\left(3x\right)+\frac{223}{81}x\sin\left(3x\right)+\frac{10}{27}x\cos\left(3x\right)+0.917695\cos\left(3x\right)-\frac{10}{81}\sin\left(3x\right)+C_0$
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Step-by-step Solution

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Multiplying the fraction by $\sin\left(3x\right)$

$\int\frac{\left(x^4+20x^3+150x^2\right)\sin\left(3x\right)}{12}dx$

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

$\int\frac{\left(x^4+20x^3+150x^2\right)\sin\left(3x\right)}{12}dx$

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Learn how to solve integral calculus problems step by step online. Find the integral int((x^4+20x^3150x^2)/12sin(3x))dx. Multiplying the fraction by \sin\left(3x\right). Take the constant \frac{1}{12} out of the integral. Divide 1 by 12. We can solve the integral \int\left(x^4+20x^3+150x^2\right)\sin\left(3x\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).

Final Answer

$-\frac{1}{36}x^4\cos\left(3x\right)-\frac{5}{9}x^3\cos\left(3x\right)-\frac{223}{54}x^{2}\cos\left(3x\right)+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{5}{9}x^{2}\sin\left(3x\right)+\frac{223}{81}x\sin\left(3x\right)+\frac{10}{27}x\cos\left(3x\right)+0.917695\cos\left(3x\right)-\frac{10}{81}\sin\left(3x\right)+C_0$

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 (x^4+20x^3)/12sin3xdx using basic integralsSolve integral of (x^4+20x^3)/12sin3xdx using u-substitutionSolve integral of (x^4+20x^3)/12sin3xdx using integration by partsSolve integral of (x^4+20x^3)/12sin3xdx using tabular integration

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Function Plot

Plotting: $-\frac{1}{36}x^4\cos\left(3x\right)-\frac{5}{9}x^3\cos\left(3x\right)-\frac{223}{54}x^{2}\cos\left(3x\right)+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{5}{9}x^{2}\sin\left(3x\right)+\frac{223}{81}x\sin\left(3x\right)+\frac{10}{27}x\cos\left(3x\right)+0.917695\cos\left(3x\right)-\frac{10}{81}\sin\left(3x\right)+C_0$

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0
a
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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

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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.

Used Formulas

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