Is reduction of order a simpler method for solving this integral?

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In summary, the formula for calculating the integral of e^t / t is ∫(e^t / t) dt = ln|t| + C, and it can be solved analytically. However, there are special cases where it cannot be solved, such as when the limits of integration include t = 0. In such cases, numerical methods, such as the trapezoidal rule or Simpson's rule, can be used to solve the integral. In real-world applications, the integral of e^t / t is used in various fields of science and engineering to model growth or decay processes and in calculating areas under curves for statistical and probabilistic purposes.
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VonWeber
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The problem has ty'' - (1 + t)y' + y = (t^2)e^2t

y1 = 1 + t

Solve by reduction of order

When I solve by variation of parameters I get:

y = .5te^2t - .5e^2t + ce^t + d(1 + t)

But solving with reduction of order gives very difficult integrals
 

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Well I should have seen this one

(ue^u - e^u)/u^2 is a quotient rule.
 

FAQ: Is reduction of order a simpler method for solving this integral?

What is the formula for calculating the integral of e^t / t?

The formula for calculating the integral of e^t / t is ∫(e^t / t) dt = ln|t| + C, where C is a constant.

Is the integral of e^t / t possible to solve analytically?

Yes, the integral of e^t / t can be solved analytically using the formula ∫(e^t / t) dt = ln|t| + C.

Are there any special cases in which the integral of e^t / t cannot be solved?

Yes, the integral of e^t / t cannot be solved when the limits of integration include t = 0, as the function is undefined at that point.

Can the integral of e^t / t be solved using numerical methods?

Yes, the integral of e^t / t can be solved using numerical methods such as the trapezoidal rule or Simpson's rule.

How is the integral of e^t / t used in real-world applications?

The integral of e^t / t is used in various fields of science and engineering, such as physics, chemistry, and economics, to model growth or decay processes. It is also used in calculating the area under a curve, which has applications in statistics and probability.

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