Integrate the rational function \(\dfrac{x}{(x+1)(x+2)}\).
\(\log\left|\dfrac{(x+2)^2}{x+1}\right|+C\)
Integrate the rational function \(\dfrac{1}{x^2-9}\).
\(\dfrac{1}{6}\log\left|\dfrac{x-3}{x+3}\right|+C\)
Integrate the rational function \(\dfrac{3x-1}{(x-1)(x-2)(x-3)}\).
\(\log|x-1|-5\log|x-2|+4\log|x-3|+C\)
Integrate the rational function \(\dfrac{x}{(x-1)(x-2)(x-3)}\).
\(\dfrac{1}{2}\log|x-1|-2\log|x-2|+\dfrac{3}{2}\log|x-3|+C\)
Integrate the rational function \(\dfrac{2x}{x^2+3x+2}\).
\(4\log|x+2|-2\log|x+1|+C\)
Integrate the rational function \(\dfrac{1-x^2}{x(1-2x)}\).
\(\dfrac{x}{2}+\log|x|-\dfrac{3}{4}\log|1-2x|+C\)
Integrate the rational function \(\dfrac{x}{(x^2+1)(x-1)}\).
\(\dfrac{1}{2}\log|x-1|-\dfrac{1}{4}\log(x^2+1)+\dfrac{1}{2}\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{x}{(x-1)^2(x+2)}\).
\(\dfrac{2}{9}\log\left|\dfrac{x-1}{x+2}\right|-\dfrac{1}{3(x-1)}+C\)
Integrate the rational function \(\dfrac{3x+5}{x^3-x^2-x+1}\).
\(\dfrac{1}{2}\log\left|\dfrac{x+1}{x-1}\right|-\dfrac{4}{x-1}+C\)
Integrate the rational function \(\dfrac{2x-3}{(x^2-1)(2x+3)}\).
\(\dfrac{5}{2}\log|x+1|-\dfrac{1}{10}\log|x-1|-\dfrac{12}{5}\log|2x+3|+C\)
Integrate the rational function \(\dfrac{5x}{(x+1)(x^2-4)}\).
\(\dfrac{5}{3}\log|x+1|-\dfrac{5}{2}\log|x+2|+\dfrac{5}{6}\log|x-2|+C\)
Integrate the rational function \(\dfrac{x^3+x+1}{x^2-1}\).
\(\dfrac{x^2}{2}+\dfrac{1}{2}\log|x+1|+\dfrac{3}{2}\log|x-1|+C\)
Integrate the rational function \(\dfrac{2}{(1-x)(1+x^2)}\).
\(-\log|x-1|+\dfrac{1}{2}\log(1+x^2)+\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{3x-1}{(x+2)^2}\).
\(3\log|x+2|+\dfrac{7}{x+2}+C\)
Integrate the rational function \(\dfrac{1}{x^4-1}\).
\(\dfrac{1}{4}\log\left|\dfrac{x-1}{x+1}\right|-\dfrac{1}{2}\tan^{-1}x+C\)
Integrate the rational function \(\dfrac{1}{x(x^n+1)}\).
Hint: multiply numerator and denominator by \(x^{n-1}\) and put \(x^n=t\).
\(\dfrac{1}{n}\log\left|\dfrac{x^n}{x^n+1}\right|+C\)
Integrate \(\dfrac{\cos x}{(1-\sin x)(2-\sin x)}\).
Hint: Put \(\sin x=t\).
\(\log\left|\dfrac{2-\sin x}{1-\sin x}\right|+C\)
Integrate the rational function \(\dfrac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}\).
\(x+\dfrac{2}{\sqrt{3}}\tan^{-1}\left(\dfrac{x}{\sqrt{3}}\right)-3\tan^{-1}\left(\dfrac{x}{2}\right)+C\)
Integrate the rational function \(\dfrac{2x}{(x^2+1)(x^2+3)}\).
\(\dfrac{1}{2}\log\left(\dfrac{x^2+1}{x^2+3}\right)+C\)
Integrate the rational function \(\dfrac{1}{x(x^4-1)}\).
\(\dfrac{1}{4}\log\left|\dfrac{x^4-1}{x^4}\right|+C\)
Integrate \(\dfrac{1}{e^x-1}\).
Hint: Put \(e^x=t\).
\(\log\left(\dfrac{e^x-1}{e^x}\right)+C\)
Choose the correct answer:
\(\displaystyle \int \dfrac{x\,dx}{(x-1)(x-2)}\) equals
(A) \(\log\left|\dfrac{(x-1)^2}{x-2}\right|+C\)
(B) \(\log\left|\dfrac{(x-2)^2}{x-1}\right|+C\)
(C) \(\log\left|\left(\dfrac{x-1}{x-2}\right)^2\right|+C\)
(D) \(\log|(x-1)(x-2)|+C\)
B
Choose the correct answer:
\(\displaystyle \int \dfrac{dx}{x(x^2+1)}\) equals
(A) \(\log|x|-\dfrac{1}{2}\log(x^2+1)+C\)
(B) \(\log|x|+\dfrac{1}{2}\log(x^2+1)+C\)
(C) \(-\log|x|+\dfrac{1}{2}\log(x^2+1)+C\)
(D) \(\dfrac{1}{2}\log|x|+\log(x^2+1)+C\)
A