Integrals Involving Exponential and Logarithmic Functions

Exponential integrands

Logarithmic integrands

The decay and half-life computation

The logarithm's own integral

Which expression is the indefinite integral of e^(ax) dx for nonzero constant a?

Which integral matches the u prime over u pattern and evaluates to a logarithm?

A radioactive isotope decays as m(t) = m_0 e^(-kt). Which equation gives its half-life T?

A population grows continuously at rate k from initial P_0. What integral gives the total population added from time 0 to T?

The integral from 0 to 1 of e^(2x) dx equals (e^2 - 1) over a constant. What constant? Answer with a number.

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?

The integral of (ln x)/x dx uses u = ln x. What is the result? Answer like (ln x)^2/2 + C.

The integral of 1/(x ln x) dx equals ln of what expression, plus C? Answer like ln|ln x|.

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?

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.