8.01SC | Fall 2016 | Undergraduate
Classical Mechanics
Week 1: Kinematics

2.5 List of Useful Integrals

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List of useful integrals

  • Integral of a polynomial function:

    If \(x(t)=At^n \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{n+1}(t_{f}^{n+1}-t_{i}^{n+1}) \)

    where \(A\) and \(n\) are constants.

  • Integral of an exponential function:

    If \(x(t)=A e^{bt}\Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{b}(e^{bt_{f}}-e^{bt_{i}}) \)

    where \(A\) and \(b\) are constants.

  • Integral of \(1/t\):

    If \(x(t)=\frac{1}{t} \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\ln(t_{f})-\ln(t_{i})=\ln(\frac{t_{f}}{t_{i}}) \)

  • Integral of sine:

    If \(x(t)=A\sin(b+ct) \Longrightarrow \int x(t)dt=-\frac{A}{c}\cos(b+ct) + D \)

    where \(A\), \(b\) and \(c\) are constants, and \(D\) is an integration constant.

  • Integral of cosine:

    If \(x(t)=A\cos(b+ct) \Longrightarrow \int x(t)dt=\frac{A}{c} \sin(b+ct) + D \)

    where \(A\), \(b\) and \(c\) are constants, and \(D\) is an integration constant.

External References

Wolfram Alpha

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