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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