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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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Solution

Final answer to the problem

$\frac{\left(-x^4-20x^3-150x^2\right)\cos\left(3x\right)}{36}+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{1}{108}\left(60x^{2}+300x\right)\sin\left(3x\right)-\frac{1}{9}\left(-\frac{1}{3}\left(x+5\right)^{2}\cos\left(3x\right)+\frac{2}{3}\left(\left(\frac{1}{3}x+\frac{5}{3}\right)\sin\left(3x\right)+\frac{1}{9}\cos\left(3x\right)\right)\right)+C_0$
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Step-by-step Solution

How should I solve this problem?

  • Integrate by parts
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
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We can solve the integral $\int\frac{x^4+20x^3+150x^2}{12}\sin\left(3x\right)dx$ 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 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 problems step by step online. Find the integral int((x^4+20x^3150x^2)/12sin(3x))dx. We can solve the integral \int\frac{x^4+20x^3+150x^2}{12}\sin\left(3x\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

$\frac{\left(-x^4-20x^3-150x^2\right)\cos\left(3x\right)}{36}+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{1}{108}\left(60x^{2}+300x\right)\sin\left(3x\right)-\frac{1}{9}\left(-\frac{1}{3}\left(x+5\right)^{2}\cos\left(3x\right)+\frac{2}{3}\left(\left(\frac{1}{3}x+\frac{5}{3}\right)\sin\left(3x\right)+\frac{1}{9}\cos\left(3x\right)\right)\right)+C_0$

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Function Plot

Plotting: $\frac{\left(-x^4-20x^3-150x^2\right)\cos\left(3x\right)}{36}+\frac{1}{27}x^{3}\sin\left(3x\right)+\frac{1}{108}\left(60x^{2}+300x\right)\sin\left(3x\right)-\frac{1}{9}\left(-\frac{1}{3}\left(x+5\right)^{2}\cos\left(3x\right)+\frac{2}{3}\left(\left(\frac{1}{3}x+\frac{5}{3}\right)\sin\left(3x\right)+\frac{1}{9}\cos\left(3x\right)\right)\right)+C_0$

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∞
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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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