Calculus Volume 1 · 免费预览
含指数函数与对数函数的积分
指数型被积函数采用换元法：或者指数中已出现其导数，或者通过处理构造出该导数；对数型被积函数则可归结为倒数模式。这些形式可用来处理增长模型和对数尺度。
1. Exponential integrands
Exponential integrands
2. Logarithmic integrands
Logarithmic integrands
3. The decay and half-life computation
The decay and half-life computation
4. The logarithm's own integral
The logarithm's own integral
5. Linear exponential formula
Which expression is the indefinite integral of e^(ax) dx for nonzero constant a?
6. Reciprocal pattern signature
Which integral matches the u prime over u pattern and evaluates to a logarithm?
7. Half-life relation
A radioactive isotope decays as m(t) = m_0 e^(-kt). Which equation gives its half-life T?
8. Growth accumulation
A population grows continuously at rate k from initial P_0. What integral gives the total population added from time 0 to T?
9. Exponential definite value
The integral from 0 to 1 of e^(2x) dx equals (e^2 - 1) over a constant. What constant? Answer with a number.
10. Which constant divides
The integral of e^(kx) dx for nonzero k equals e^(kx) over k plus C. For k = -3, what is the integral of e^(-3x) dx?
11. Log integral with x## practice-fill Log integral with x
The integral of (ln x)/x dx uses u = ln x. What is the result? Answer like (ln x)^2/2 + C.
12. Double logarithm
The integral of 1/(x ln x) dx equals ln of what expression, plus C? Answer like ln|ln x|.
13. Continuous compounding stream
Money flows into an account at 1000 e^(0.05t) dollars per year under continuous compounding at 5 percent. Which integral gives the accumulated deposits over the first 10 years?
14. Decayed fraction## practice-fill Decayed fraction
In decay m(t) = m_0 e^(-kt), what fraction of the original mass remains at the half-life T? Answer with a fraction like 1/3.