allery$\displaystyle\int_0^{\pi/2} \log(\tan x)\, dx = ?$
A$-(\pi/2)\ln 2$
B$0$ (by King's symmetry between $\sin$ and $\cos$)
C$\pi\ln 2$
D$\ln 2$
Answer & Solution
Correct answer: B. $0$ (by King's symmetry between $\sin$ and $\cos$)
$I = \int \log\tan x = \int(\log\sin x - \log\cos x)$.
Both pieces equal by King's (previous problem), so the difference is 0.
Keep practising Unit 6A: Indefinite Integrals & Fundamental Theorem
Browse all Unit 6A: Indefinite Integrals & Fundamental Theorem questions for AP Calculus AB, or explore more Mathematics topics.
Related questions
The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} The number of arbitrary constants in the general solution of a differential equation of orUsing the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:$\displaystyle\int \log x\,dx$ equals:$\displaystyle\int ec^2 x\,dx$ equals:$\displaystyle\int \dfrac{1}{ qrt{1 - x^2}}\,dx$ equals:$\displaystyle\int \dfrac{1}{1+x^2}\,dx$ equals: