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

Used Formulas

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Basic Derivatives

· Sum Rule for Differentiation
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)$
$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Power rule for derivatives
$\frac{d}{dx}\left(x^a\right)=ax^{\left(a-1\right)}$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$
· Derivative of a Constant
$\frac{d}{dx}\left(c\right)=0$

Integration Techniques

· Integration by Substitution
$\int f\left(x\right)dx=\int f\left(g\left(t\right)\right) g'\left(t\right)dt$

Trigonometric Integrals

· Integral of the sine function
$\int\sin\left(\theta \right)dx=-\cos\left(\theta \right)+C$
$\int\cos\left(ax\right)dx=\frac{1}{a}\sin\left(ax\right)+C$
$\int\sin\left(ax\right)dx=-\left(\frac{1}{a}\right)\cos\left(ax\right)+C$

Basic Integrals

· Constant factor Rule
$\int cxdx=c\int xdx$

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

See formulas (10)

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