Indefinite Integrals with hyperbolic cotangent (coth x)
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| 28 Integrals of coth ax | |
|---|---|
| $ \int coth ax dx=\dfrac{\ln sh ax}{a} +C $ | |
| $ \int coth^{2} ax dx=x-\dfrac{coth ax}{a} +C $ | |
| $ \int coth^{3} ax dx=\dfrac{1}{a}\dfrac{\ln sh ax}{a}-\dfrac{coth^{2} ax}{2a} +C $ | |
| $ \int\dfrac{coth^{n} ax}{sh^{2} ax} dx=\dfrac{coth^{n+1} ax}{(n+1)a} +C $ | |
| $ \int\dfrac{dx}{coth ax sh^{2} ax} dx=\dfrac{1}{a}\ln coth ax +C $ | |
| $ \int\dfrac{dx}{coth ax} dx=\dfrac{1}{a}\ln ch ax +C $ | |
| $ \int x coth ax dx=\dfrac{1}{a^{2}}\biggl\{ ax+\dfrac{(ax)^{3}}{9}-\dfrac{(ax)^{5}}{225}+\dfrac{2(ax)^{7}}{105}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} +C $ | |
| $ \int x coth^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x coth ax}{a}+\dfrac{1}{a^{2}}\ln sh ax+C $ | |
| $ \int\dfrac{coth ax}{x} dx=\biggl\{-\dfrac{1}{ax}+\dfrac{ax}{3}-\dfrac{(ax)^{3}}{135}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} +C $ | |
| $ \int\dfrac{dx}{p+q coth ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(p sh ax+q ch ax) +C $ | |
| $ \int coth^{n} ax dx=-\dfrac{coth^{n-1} ax}{a(n-1)}+ \int coth^{n-2} ax dx $ | |
