- List of integrals of hyperbolic functions
The following is a list of
integral s (antiderivative functions) ofhyperbolic function s. For a complete list of Integral functions, seelist of integrals .In all formulas the constant "a" is assumed to be nonzero, and "C"denotes the
constant of integration .: intsinh ax,dx = frac{1}{a}cosh ax+C,
: intcosh ax,dx = frac{1}{a}sinh ax+C,
: intsinh^2 ax,dx = frac{1}{4a}sinh 2ax - frac{x}{2}+C,
: intcosh^2 ax,dx = frac{1}{4a}sinh 2ax + frac{x}{2}+C,
: int anh^2 ax,dx = x - frac{ anh ax}{a}+C,
: intsinh^n ax,dx = frac{1}{an}sinh^{n-1} axcosh ax - frac{n-1}{n}intsinh^{n-2} ax,dx qquadmbox{(for }n>0mbox{)},
: also: intsinh^n ax,dx = frac{1}{a(n+1)}sinh^{n+1} axcosh ax - frac{n+2}{n+1}intsinh^{n+2}ax,dx qquadmbox{(for }n<0mbox{, }n eq -1mbox{)},
: intcosh^n ax,dx = frac{1}{an}sinh axcosh^{n-1} ax + frac{n-1}{n}intcosh^{n-2} ax,dx qquadmbox{(for }n>0mbox{)},
: also: intcosh^n ax,dx = -frac{1}{a(n+1)}sinh axcosh^{n+1} ax - frac{n+2}{n+1}intcosh^{n+2}ax,dx qquadmbox{(for }n<0mbox{, }n eq -1mbox{)},
: intfrac{dx}{sinh ax} = frac{1}{a} lnleft| anhfrac{ax}{2} ight|+C,
: also: intfrac{dx}{sinh ax} = frac{1}{a} lnleft|frac{cosh ax - 1}{sinh ax} ight|+C,
: also: intfrac{dx}{sinh ax} = frac{1}{a} lnleft|frac{sinh ax}{cosh ax + 1} ight|+C,
: also: intfrac{dx}{sinh ax} = frac{1}{a} lnleft|frac{cosh ax - 1}{cosh ax + 1} ight|+C,
: intfrac{dx}{cosh ax} = frac{2}{a} arctan e^{ax}+C,
: intfrac{dx}{sinh^n ax} = frac{cosh ax}{a(n-1)sinh^{n-1} ax}-frac{n-2}{n-1}intfrac{dx}{sinh^{n-2} ax} qquadmbox{(for }n eq 1mbox{)},
: intfrac{dx}{cosh^n ax} = frac{sinh ax}{a(n-1)cosh^{n-1} ax}+frac{n-2}{n-1}intfrac{dx}{cosh^{n-2} ax} qquadmbox{(for }n eq 1mbox{)},
: intfrac{cosh^n ax}{sinh^m ax} dx = frac{cosh^{n-1} ax}{a(n-m)sinh^{m-1} ax} + frac{n-1}{n-m}intfrac{cosh^{n-2} ax}{sinh^m ax} dx qquadmbox{(for }m eq nmbox{)},
: also: intfrac{cosh^n ax}{sinh^m ax} dx = -frac{cosh^{n+1} ax}{a(m-1)sinh^{m-1} ax} + frac{n-m+2}{m-1}intfrac{cosh^n ax}{sinh^{m-2} ax} dx qquadmbox{(for }m eq 1mbox{)},
: also: intfrac{cosh^n ax}{sinh^m ax} dx = -frac{cosh^{n-1} ax}{a(m-1)sinh^{m-1} ax} + frac{n-1}{m-1}intfrac{cosh^{n-2} ax}{sinh^{m-2} ax} dx qquadmbox{(for }m eq 1mbox{)},
: intfrac{sinh^m ax}{cosh^n ax} dx = frac{sinh^{m-1} ax}{a(m-n)cosh^{n-1} ax} + frac{m-1}{m-n}intfrac{sinh^{m-2} ax}{cosh^n ax} dx qquadmbox{(for }m eq nmbox{)},
: also: intfrac{sinh^m ax}{cosh^n ax} dx = frac{sinh^{m+1} ax}{a(n-1)cosh^{n-1} ax} + frac{m-n+2}{n-1}intfrac{sinh^m ax}{cosh^{n-2} ax} dx qquadmbox{(for }n eq 1mbox{)},
: also: intfrac{sinh^m ax}{cosh^n ax} dx = -frac{sinh^{m-1} ax}{a(n-1)cosh^{n-1} ax} + frac{m-1}{n-1}intfrac{sinh^{m -2} ax}{cosh^{n-2} ax} dx qquadmbox{(for }n eq 1mbox{)},
: int xsinh ax,dx = frac{1}{a} xcosh ax - frac{1}{a^2}sinh ax+C,
: int xcosh ax,dx = frac{1}{a} xsinh ax - frac{1}{a^2}cosh ax+C,
: int x^2 cosh ax,dx = -frac{2x cosh ax}{a^2} + left(frac{x^2}{a}+frac{2}{a^3} ight) sinh ax+C,
: int anh ax,dx = frac{1}{a}ln|cosh ax|+C,
: int coth ax,dx = frac{1}{a}ln|sinh ax|+C,
: int anh^n ax,dx = -frac{1}{a(n-1)} anh^{n-1} ax+int anh^{n-2} ax,dx qquadmbox{(for }n eq 1mbox{)},
: int coth^n ax,dx = -frac{1}{a(n-1)}coth^{n-1} ax+intcoth^{n-2} ax,dx qquadmbox{(for }n eq 1mbox{)},
: int sinh ax sinh bx,dx = frac{1}{a^2-b^2} (asinh bx cosh ax - bcosh bx sinh ax)+C qquadmbox{(for }a^2 eq b^2mbox{)},
: int cosh ax cosh bx,dx = frac{1}{a^2-b^2} (asinh ax cosh bx - bsinh bx cosh ax)+C qquadmbox{(for }a^2 eq b^2mbox{)},
: int cosh ax sinh bx,dx = frac{1}{a^2-b^2} (asinh ax sinh bx - bcosh ax cosh bx)+C qquadmbox{(for }a^2 eq b^2mbox{)},
: int sinh (ax+b)sin (cx+d),dx = frac{a}{a^2+c^2}cosh(ax+b)sin(cx+d)-frac{c}{a^2+c^2}sinh(ax+b)cos(cx+d)+C,
: int sinh (ax+b)cos (cx+d),dx = frac{a}{a^2+c^2}cosh(ax+b)cos(cx+d)+frac{c}{a^2+c^2}sinh(ax+b)sin(cx+d)+C,
: int cosh (ax+b)sin (cx+d),dx = frac{a}{a^2+c^2}sinh(ax+b)sin(cx+d)-frac{c}{a^2+c^2}cosh(ax+b)cos(cx+d)+C,
: int cosh (ax+b)cos (cx+d),dx = frac{a}{a^2+c^2}sinh(ax+b)cos(cx+d)+frac{c}{a^2+c^2}cosh(ax+b)sin(cx+d)+C,
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