What is the meaning of Si in the integral of ln(x)cos(x)?

In summary, we can use integration by parts to solve the integral of ln(x).cos(x). This results in the equation ln(x).sin(x) - Si(x) + c, where Si(x) is the integral of sin(x)/x. This can be represented as (2) and can be used to continue solving the integral.
  • #1
leprofece
241
0
integral ln(x).cos(x)
Here I have some clear ideas
U = lnx du = 1/x
dv = cosx so int de cosx = v = -sinx
-sinxlnx -int (sinx)/(x)

Ok I think I must integrate again
u= sinx du = cosx
dv = 1/x v = lnx
Again I got -sinxlnx -int (sinx lnx)
But I am stuck here and I don't know how to finish it??
Can you help me?

Ok i found that integral of sinx/x is Si according to a program but what does Si mean?
 
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  • #2
leprofece said:
integral ln(x).cos(x)
Here I have some clear ideas
U = lnx du = 1/x
dv = cosx so int de cosx = v = -sinx
-sinxlnx -int (sinx)/(x)

Ok I think I must integrate again
u= sinx du = cosx
dv = 1/x v = lnx
Again I got -sinxlnx -int (sinx lnx)
But I am stuck here and I don't know how to finish it??
Can you help me?

Ok i found that integral of sinx/x is Si according to a program but what does Si mean?

Integrating by parts You obtain...$\displaystyle \int \ln x\ \cos x\ d x = \ln x\ \sin x - \int \frac{\sin x} {x}\ d x = \ln x\ \sin x - \text{Si}\ (x) + c\ (1)$

... where...

$\displaystyle \text{Si}\ (x) = \int_{0}^{x} \frac{\sin t}{t}\ d t\ (2)$

Kind regards

$\chi$ $\sigma$
 

FAQ: What is the meaning of Si in the integral of ln(x)cos(x)?

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is the inverse operation of differentiation and is used to find the original function when its derivative is known.

2. How do you solve an integral?

To solve an integral, you need to use integration techniques such as u-substitution, integration by parts, or trigonometric substitution. It involves finding an antiderivative of the integrand and evaluating it at the upper and lower limits of integration.

3. What is ln(x)cos(x)?

ln(x)cos(x) is a function that is the product of the natural logarithm of x and the cosine of x. It is commonly used in calculus and can represent various physical phenomena, such as the growth of a population or the oscillation of a spring.

4. Why is it important to solve integrals?

Solving integrals is important in various fields of science and engineering, such as physics, chemistry, and economics. It allows us to calculate quantities such as area, volume, and displacement, which are crucial for understanding and predicting real-world phenomena.

5. What is the result of solving the integral ln(x)cos(x)?

The result of solving the integral ln(x)cos(x) will depend on the limits of integration. Generally, the integral will result in a combination of logarithmic and trigonometric functions. For example, the result of integrating ln(x)cos(x) from 1 to e will be 1.

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