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Find the integral $\int e^{\frac{-5x}{2}}\sin\left(\frac{\sqrt{3}}{2}x\right)dx$

Used Formulas

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Solution

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Derivatives of trigonometric functions

· Derivative of the sine function
$\frac{d}{dx}\left(\sin\left(\theta \right)\right)=\frac{d}{dx}\left(\theta \right)\cos\left(\theta \right)$
· Derivative of the cosine function
$\frac{d}{dx}\left(\cos\left(\theta \right)\right)=-\frac{d}{dx}\left(\theta \right)\sin\left(\theta \right)$

Basic Derivatives

$\frac{d}{dx}\left(\frac{x}{c}\right)=\frac{1}{c}\frac{d}{dx}\left(x\right)$
· Product rule for derivatives
$\frac{d}{dx}\left(ab\right)=\frac{d}{dx}\left(a\right)b+a\frac{d}{dx}\left(b\right)$
· Derivative of a Constant
$\frac{d}{dx}\left(c\right)=0$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$

Integration Techniques

· Integration by Substitution
$\int f\left(x\right)dx=\int f\left(g\left(t\right)\right) g'\left(t\right)dt$
· Integration by Parts
$\int udv=uv - \int vdu$

Basic Integrals

· Constant factor Rule
$\int cxdx=c\int xdx$
$\int e^xdx=e^x+C$

Similar Problems Solved

Find the integral

$\int e^{3x}\cdot\cos\left(4x\right)dx$

Find the integral

$\int e^xcos2xdx$

Find the integral

$\int e^{3t}sen2tdt$

Find the integral

$\int\left(e^{2x}\cos\left(5x\right)\right)dx$

Find the integral

$\int x^2e^{2x+1}dx$

Find the integral

$\int e^{2x}cosxdx$

Find the integral

$\int e^{x}\cos xdx$

Main Topic: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

Used Formulas

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