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Este artigo ou seção está a ser traduzido (desde agosto de 2007). Ajude e colabore com a tradução.

A lista seguinte contém integrais de funções trigonométricas.

A constante "c" é assumida como não nula.

Integrais de funções trigonométricas contendo apenas seno |

∫sen⁡cxdx=−1ccos⁡cx{displaystyle int operatorname {sen} cx;dx=-{frac {1}{c}}cos cx}

∫senn⁡cxdx=−senn−1⁡cxcos⁡cxnc+n−1n∫senn−2⁡cxdx(for n>0){displaystyle int operatorname {sen} ^{n}cx;dx=-{frac {operatorname {sen} ^{n-1}cxcos cx}{nc}}+{frac {n-1}{n}}int operatorname {sen} ^{n-2}cx;dxqquad {mbox{(for }}n>0{mbox{)}}}

∫1−sen⁡xdx=∫cvs⁡xdx=2cos⁡x2+sen⁡x2cos⁡x2−sen⁡x2cvs⁡x{displaystyle int {sqrt {1-operatorname {sen} {x}}},dx=int {sqrt {operatorname {cvs} {x}}},dx=2{frac {cos {frac {x}{2}}+operatorname {sen} {frac {x}{2}}}{cos {frac {x}{2}}-operatorname {sen} {frac {x}{2}}}}{sqrt {operatorname {cvs} {x}}}}

onde cvs{x} é a função de Coversene

∫xsen⁡cxdx=sen⁡cxc2−xcos⁡cxc{displaystyle int xoperatorname {sen} cx;dx={frac {operatorname {sen} cx}{c^{2}}}-{frac {xcos cx}{c}}}

∫xnsen⁡cxdx=−xnccos⁡cx+nc∫xn−1cos⁡cxdx(for n>0){displaystyle int x^{n}operatorname {sen} cx;dx=-{frac {x^{n}}{c}}cos cx+{frac {n}{c}}int x^{n-1}cos cx;dxqquad {mbox{(for }}n>0{mbox{)}}}

∫sen⁡cxxdx=∑i=0∞(−1)i(cx)2i+1(2i+1)⋅(2i+1)!{displaystyle int {frac {operatorname {sen} cx}{x}}dx=sum _{i=0}^{infty }(-1)^{i}{frac {(cx)^{2i+1}}{(2i+1)cdot (2i+1)!}}}

∫sen⁡cxxndx=−sin⁡cx(n−1)xn−1+cn−1∫cos⁡cxxn−1dx{displaystyle int {frac {operatorname {sen} cx}{x^{n}}}dx=-{frac {sin cx}{(n-1)x^{n-1}}}+{frac {c}{n-1}}int {frac {cos cx}{x^{n-1}}}dx}

∫dxsen⁡cx=1cln⁡|tan⁡cx2|{displaystyle int {frac {dx}{operatorname {sen} cx}}={frac {1}{c}}ln left|tan {frac {cx}{2}}right|}

∫dxsenn⁡cx=cos⁡cxc(1−n)senn−1⁡cx+n−2n−1∫dxsenn−2⁡cx(for n>1){displaystyle int {frac {dx}{operatorname {sen} ^{n}cx}}={frac {cos cx}{c(1-n)operatorname {sen} ^{n-1}cx}}+{frac {n-2}{n-1}}int {frac {dx}{operatorname {sen} ^{n-2}cx}}qquad {mbox{(for }}n>1{mbox{)}}}

∫dx1±sen⁡cx=1ctan⁡(cx2∓π4){displaystyle int {frac {dx}{1pm operatorname {sen} cx}}={frac {1}{c}}tan left({frac {cx}{2}}mp {frac {pi }{4}}right)}

∫xdx1+sen⁡cx=xctan⁡(cx2−π4)+2c2ln⁡|cos⁡(cx2−π4)|{displaystyle int {frac {x;dx}{1+operatorname {sen} cx}}={frac {x}{c}}tan left({frac {cx}{2}}-{frac {pi }{4}}right)+{frac {2}{c^{2}}}ln left|cos left({frac {cx}{2}}-{frac {pi }{4}}right)right|}

∫xdx1−sen⁡cx=xccot⁡(π4−cx2)+2c2ln⁡|sen⁡(π4−cx2)|{displaystyle int {frac {x;dx}{1-operatorname {sen} cx}}={frac {x}{c}}cot left({frac {pi }{4}}-{frac {cx}{2}}right)+{frac {2}{c^{2}}}ln left|operatorname {sen} left({frac {pi }{4}}-{frac {cx}{2}}right)right|}

∫sen⁡cxdx1±sen⁡cx=±x+1ctan⁡(π4∓cx2){displaystyle int {frac {operatorname {sen} cx;dx}{1pm operatorname {sen} cx}}=pm x+{frac {1}{c}}tan left({frac {pi }{4}}mp {frac {cx}{2}}right)}

∫sen⁡c1xsen⁡c2xdx=sen⁡(c1−c2)x2(c1−c2)−sen⁡(c1+c2)x2(c1+c2)(for |c1|≠|c2|){displaystyle int operatorname {sen} c_{1}xoperatorname {sen} c_{2}x;dx={frac {operatorname {sen} (c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{frac {operatorname {sen} (c_{1}+c_{2})x}{2(c_{1}+c_{2})}}qquad {mbox{(for }}|c_{1}|neq |c_{2}|{mbox{)}}}

Integrais de funções trigonométricas contendo apenas cosseno |

∫cos⁡cxdx=1csen⁡cx{displaystyle int cos cx;dx={frac {1}{c}}operatorname {sen} cx}

∫cosn⁡cxdx=cosn−1⁡cxsen⁡cxnc+n−1n∫cosn−2⁡cxdx(para n>0){displaystyle int cos ^{n}cx;dx={frac {cos ^{n-1}cxoperatorname {sen} cx}{nc}}+{frac {n-1}{n}}int cos ^{n-2}cx;dxqquad {mbox{(para }}n>0{mbox{)}}}

∫xcos⁡cxdx=cos⁡cxc2+xsen⁡cxc{displaystyle int xcos cx;dx={frac {cos cx}{c^{2}}}+{frac {xoperatorname {sen} cx}{c}}}

CALC

∫xncos⁡cxdx=xnsin⁡cxc−nc∫xn−1sin⁡cxdx{displaystyle int x^{n}cos cx;dx={frac {x^{n}sin cx}{c}}-{frac {n}{c}}int x^{n-1}sin cx;dx}

∫cos⁡cxxdx=ln⁡|cx|+∑i=1∞(−1)i(cx)2i2i⋅(2i)!{displaystyle int {frac {cos cx}{x}}dx=ln |cx|+sum _{i=1}^{infty }(-1)^{i}{frac {(cx)^{2i}}{2icdot (2i)!}}}

∫cos⁡cxxndx=−cos⁡cx(n−1)xn−1−cn−1∫sen⁡cxxn−1dx(for n≠1){displaystyle int {frac {cos cx}{x^{n}}}dx=-{frac {cos cx}{(n-1)x^{n-1}}}-{frac {c}{n-1}}int {frac {operatorname {sen} cx}{x^{n-1}}}dxqquad {mbox{(for }}nneq 1{mbox{)}}}

∫dxcos⁡cx=1cln⁡|tan⁡(cx2+π4)|{displaystyle int {frac {dx}{cos cx}}={frac {1}{c}}ln left|tan left({frac {cx}{2}}+{frac {pi }{4}}right)right|}

∫dxcosn⁡cx=sen⁡cxc(n−1)cosn−1cx+n−2n−1∫dxcosn−2⁡cx(for n>1){displaystyle int {frac {dx}{cos ^{n}cx}}={frac {operatorname {sen} cx}{c(n-1)cos^{n-1}cx}}+{frac {n-2}{n-1}}int {frac {dx}{cos ^{n-2}cx}}qquad {mbox{(for }}n>1{mbox{)}}}

∫dx1+cos⁡cx=1ctan⁡cx2{displaystyle int {frac {dx}{1+cos cx}}={frac {1}{c}}tan {frac {cx}{2}}}

∫dx1−cos⁡cx=−1ccot⁡cx2{displaystyle int {frac {dx}{1-cos cx}}=-{frac {1}{c}}cot {frac {cx}{2}}}

∫xdx1+cos⁡cx=xctan⁡cx2+2c2ln⁡|cos⁡cx2|{displaystyle int {frac {x;dx}{1+cos cx}}={frac {x}{c}}tan {cx}{2}+{frac {2}{c^{2}}}ln left|cos {frac {cx}{2}}right|}

∫xdx1−cos⁡cx=−xxcot⁡cx2+2c2ln⁡|sin⁡cx2|{displaystyle int {frac {x;dx}{1-cos cx}}=-{frac {x}{x}}cot {cx}{2}+{frac {2}{c^{2}}}ln left|sin {frac {cx}{2}}right|}

∫cos⁡cxdx1+cos⁡cx=x−1ctan⁡cx2{displaystyle int {frac {cos cx;dx}{1+cos cx}}=x-{frac {1}{c}}tan {frac {cx}{2}}}

∫cos⁡cxdx1−cos⁡cx=−x−1ccot⁡cx2{displaystyle int {frac {cos cx;dx}{1-cos cx}}=-x-{frac {1}{c}}cot {frac {cx}{2}}}

∫cos⁡c1xcos⁡c2xdx=sin⁡(c1−c2)x2(c1−c2)+sin⁡(c1+c2)x2(c1+c2)(Para |c1|≠|c2|){displaystyle int cos c_{1}xcos c_{2}x;dx={frac {sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{frac {sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}qquad {mbox{(Para }}|c_{1}|neq |c_{2}|{mbox{)}}}

Integrais de funções trigonométricas contendo apenas tangente |

∫tan⁡cxdx=−1cln⁡|cos⁡cx|{displaystyle int tan cx;dx=-{frac {1}{c}}ln |cos cx|}

∫tann⁡cxdx=1c(n−1)tann−1⁡cx−∫tann−2⁡cxdx(for n≠1){displaystyle int tan ^{n}cx;dx={frac {1}{c(n-1)}}tan ^{n-1}cx-int tan ^{n-2}cx;dxqquad {mbox{(for }}nneq 1{mbox{)}}}

∫dxtan⁡cx+1=x2+12cln⁡|sin⁡cx+cos⁡cx|{displaystyle int {frac {dx}{tan cx+1}}={frac {x}{2}}+{frac {1}{2c}}ln |sin cx+cos cx|}

∫dxtan⁡cx−1=−x2+12cln⁡|sin⁡cx−cos⁡cx|{displaystyle int {frac {dx}{tan cx-1}}=-{frac {x}{2}}+{frac {1}{2c}}ln |sin cx-cos cx|}

∫tan⁡cxdxtan⁡cx+1=x2−12cln⁡|sin⁡cx+cos⁡cx|{displaystyle int {frac {tan cx;dx}{tan cx+1}}={frac {x}{2}}-{frac {1}{2c}}ln |sin cx+cos cx|}

∫tan⁡cxdxtan⁡cx−1=x2+12cln⁡|sin⁡cx−cos⁡cx|{displaystyle int {frac {tan cx;dx}{tan cx-1}}={frac {x}{2}}+{frac {1}{2c}}ln |sin cx-cos cx|}

Integrais de funções trigonométricas contendo apenas secante |

∫sec⁡cxdx=1cln⁡|sec⁡cx+tan⁡cx|{displaystyle int sec {cx},dx={frac {1}{c}}ln {left|sec {cx}+tan {cx}right|}}

∫secn⁡cxdx=secn−1⁡cxsin⁡cxc(n−1)+n−2n−1∫secn−2⁡cxdx (for n≠1){displaystyle int sec ^{n}{cx},dx={frac {sec ^{n-1}{cx}sin {cx}}{c(n-1)}},+,{frac {n-2}{n-1}}int sec ^{n-2}{cx},dxqquad {mbox{ (for }}nneq 1{mbox{)}}}

∫dxsec⁡x+1=x−tan⁡x2{displaystyle int {frac {dx}{sec {x}+1}}=x-tan {frac {x}{2}}}

Integrais de funções trigonométricas contendo apenas cossencante |

∫csc⁡cxdx=−1cln⁡|csc⁡cx−cot⁡cx|{displaystyle int csc {cx},dx=-{frac {1}{c}}ln {left|csc {cx}-cot {cx}right|}}

∫cscn⁡cxdx=−cscn−1⁡cxcos⁡cxc(n−1)+n−2n−1∫cscn−2⁡cxdx (for n≠1){displaystyle int csc ^{n}{cx},dx=-{frac {csc ^{n-1}{cx}cos {cx}}{c(n-1)}},+,{frac {n-2}{n-1}}int csc ^{n-2}{cx},dxqquad {mbox{ (for }}nneq 1{mbox{)}}}

Integrais de funções trigonométricas contendo apenas cotangente |

∫cot⁡cxdx=1cln⁡|sin⁡cx|{displaystyle int cot cx;dx={frac {1}{c}}ln |sin cx|}

∫cotn⁡cxdx=−1c(n−1)cotn−1⁡cx−∫cotn−2⁡cxdx(for n≠1){displaystyle int cot ^{n}cx;dx=-{frac {1}{c(n-1)}}cot ^{n-1}cx-int cot ^{n-2}cx;dxqquad {mbox{(for }}nneq 1{mbox{)}}}

∫dx1+cot⁡cx=∫tan⁡cxdxtan⁡cx+1{displaystyle int {frac {dx}{1+cot cx}}=int {frac {tan cx;dx}{tan cx+1}}}

∫dx1−cot⁡cx=∫tan⁡cxdxtan⁡cx−1{displaystyle int {frac {dx}{1-cot cx}}=int {frac {tan cx;dx}{tan cx-1}}}

Integrais de funções trigonométricas contendo seno e cosseno |

∫dxcos⁡cx±sin⁡cx=1c2ln⁡|tan⁡(cx2±π8)|{displaystyle int {frac {dx}{cos cxpm sin cx}}={frac {1}{c{sqrt {2}}}}ln left|tan left({frac {cx}{2}}pm {frac {pi }{8}}right)right|}

∫dx(cos⁡cx±sin⁡cx)2=12ctan⁡(cx∓π4){displaystyle int {frac {dx}{(cos cxpm sin cx)^{2}}}={frac {1}{2c}}tan left(cxmp {frac {pi }{4}}right)}

∫dx(cos⁡x+sen⁡x)n=1n−1(sen⁡x−cos⁡x(cos⁡x+sen⁡x)n−1−2(n−2)∫dx(cos⁡x+sen⁡x)n−2){displaystyle int {frac {dx}{(cos x+operatorname {sen} x)^{n}}}={frac {1}{n-1}}left({frac {operatorname {sen} x-cos x}{(cos x+operatorname {sen} x)^{n-1}}}-2(n-2)int {frac {dx}{(cos x+operatorname {sen} x)^{n-2}}}right)}

∫cos⁡cxdxcos⁡cx+sen⁡cx=x2+12cln⁡|sen⁡cx+cos⁡cx|{displaystyle int {frac {cos cx;dx}{cos cx+operatorname {sen} cx}}={frac {x}{2}}+{frac {1}{2c}}ln left|operatorname {sen} cx+cos cxright|}

∫cos⁡cxdxcos⁡cx−sen⁡cx=x2−12cln⁡|sen⁡cx−cos⁡cx|{displaystyle int {frac {cos cx;dx}{cos cx-operatorname {sen} cx}}={frac {x}{2}}-{frac {1}{2c}}ln left|operatorname {sen} cx-cos cxright|}

∫sen⁡cxdxcos⁡cx+sen⁡cx=x2−12cln⁡|sen⁡cx+cos⁡cx|{displaystyle int {frac {operatorname {sen} cx;dx}{cos cx+operatorname {sen} cx}}={frac {x}{2}}-{frac {1}{2c}}ln left|operatorname {sen} cx+cos cxright|}

∫sen⁡cxdxcos⁡cx−sen⁡cx=−x2−12cln⁡|sen⁡cx−cos⁡cx|{displaystyle int {frac {operatorname {sen} cx;dx}{cos cx-operatorname {sen} cx}}=-{frac {x}{2}}-{frac {1}{2c}}ln left|operatorname {sen} cx-cos cxright|}

∫cos⁡cxdxsen⁡cx(1+cos⁡cx)=−14ctan2⁡cx2+12cln⁡|tan⁡cx2|{displaystyle int {frac {cos cx;dx}{operatorname {sen} cx(1+cos cx)}}=-{frac {1}{4c}}tan ^{2}{frac {cx}{2}}+{frac {1}{2c}}ln left|tan {frac {cx}{2}}right|}

∫cos⁡cxdxsen⁡cx(1+−cos⁡cx)=−14ccot2⁡cx2−12cln⁡|tan⁡cx2|{displaystyle int {frac {cos cx;dx}{operatorname {sen} cx(1+-cos cx)}}=-{frac {1}{4c}}cot ^{2}{frac {cx}{2}}-{frac {1}{2c}}ln left|tan {frac {cx}{2}}right|}

∫sin⁡cxdxcos⁡cx(1+sin⁡cx)=14ccot2⁡(cx2+π4)+12cln⁡|tan⁡(cx2+π4)|{displaystyle int {frac {sin cx;dx}{cos cx(1+sin cx)}}={frac {1}{4c}}cot ^{2}left({frac {cx}{2}}+{frac {pi }{4}}right)+{frac {1}{2c}}ln left|tan left({frac {cx}{2}}+{frac {pi }{4}}right)right|}

∫sin⁡cxdxcos⁡cx(1−sin⁡cx)=14ctan2⁡(cx2+π4)−12cln⁡|tan⁡(cx2+π4)|{displaystyle int {frac {sin cx;dx}{cos cx(1-sin cx)}}={frac {1}{4c}}tan ^{2}left({frac {cx}{2}}+{frac {pi }{4}}right)-{frac {1}{2c}}ln left|tan left({frac {cx}{2}}+{frac {pi }{4}}right)right|}

∫sin⁡cxcos⁡cxdx=12csin2⁡cx{displaystyle int sin cxcos cx;dx={frac {1}{2c}}sin ^{2}cx}

∫sin⁡c1xcos⁡c2xdx=−cos⁡(c1+c2)x2(c1+c2)−cos⁡(c1−c2)x2(c1−c2)(for |c1|≠|c2|){displaystyle int sin c_{1}xcos c_{2}x;dx=-{frac {cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{frac {cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}qquad {mbox{(for }}|c_{1}|neq |c_{2}|{mbox{)}}}

∫sinn⁡cxcos⁡cxdx=1c(n+1)sinn+1⁡cx(for n≠1){displaystyle int sin ^{n}cxcos cx;dx={frac {1}{c(n+1)}}sin ^{n+1}cxqquad {mbox{(for }}nneq 1{mbox{)}}}

∫sin⁡cxcosn⁡cxdx=−1c(n+1)cosn+1⁡cx(for n≠1){displaystyle int sin cxcos ^{n}cx;dx=-{frac {1}{c(n+1)}}cos ^{n+1}cxqquad {mbox{(for }}nneq 1{mbox{)}}}

∫sinn⁡cxcosm⁡cxdx=−sinn−1⁡cxcosm+1⁡cxc(n+m)+n−1n+m∫sinn−2⁡cxcosm⁡cxdx(for m,n>0){displaystyle int sin ^{n}cxcos ^{m}cx;dx=-{frac {sin ^{n-1}cxcos ^{m+1}cx}{c(n+m)}}+{frac {n-1}{n+m}}int sin ^{n-2}cxcos ^{m}cx;dxqquad {mbox{(for }}m,n>0{mbox{)}}}

também: ∫sinn⁡cxcosm⁡cxdx=sinn+1⁡cxcosm−1⁡cxc(n+m)+m−1n+m∫sinn⁡cxcosm−2⁡cxdx(for m,n>0){displaystyle int sin ^{n}cxcos ^{m}cx;dx={frac {sin ^{n+1}cxcos ^{m-1}cx}{c(n+m)}}+{frac {m-1}{n+m}}int sin ^{n}cxcos ^{m-2}cx;dxqquad {mbox{(for }}m,n>0{mbox{)}}}

∫dxsin⁡cxcos⁡cx=1cln⁡|tan⁡cx|{displaystyle int {frac {dx}{sin cxcos cx}}={frac {1}{c}}ln left|tan cxright|}

∫dxsin⁡cxcosn⁡cx=1c(n−1)cosn−1⁡cx+∫dxsin⁡cxcosn−2⁡cx(for n≠1){displaystyle int {frac {dx}{sin cxcos ^{n}cx}}={frac {1}{c(n-1)cos ^{n-1}cx}}+int {frac {dx}{sin cxcos ^{n-2}cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫dxsinn⁡cxcos⁡cx=−1c(n−1)sinn−1⁡cx+∫dxsinn−2⁡cxcos⁡cx(for n≠1){displaystyle int {frac {dx}{sin ^{n}cxcos cx}}=-{frac {1}{c(n-1)sin ^{n-1}cx}}+int {frac {dx}{sin ^{n-2}cxcos cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫sin⁡cxdxcosn⁡cx=1c(n−1)cosn−1⁡cx(for n≠1){displaystyle int {frac {sin cx;dx}{cos ^{n}cx}}={frac {1}{c(n-1)cos ^{n-1}cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫sin2⁡cxdxcos⁡cx=−1csin⁡cx+1cln⁡|tan⁡(π4+cx2)|{displaystyle int {frac {sin ^{2}cx;dx}{cos cx}}=-{frac {1}{c}}sin cx+{frac {1}{c}}ln left|tan left({frac {pi }{4}}+{frac {cx}{2}}right)right|}

∫sin2⁡cxdxcosn⁡cx=sin⁡cxc(n−1)cosn−1⁡cx−1n−1∫dxcosn−2⁡cx(for n≠1){displaystyle int {frac {sin ^{2}cx;dx}{cos ^{n}cx}}={frac {sin cx}{c(n-1)cos ^{n-1}cx}}-{frac {1}{n-1}}int {frac {dx}{cos ^{n-2}cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫sinn⁡cxdxcos⁡cx=−sinn−1⁡cxc(n−1)+∫sinn−2⁡cxdxcos⁡cx(for n≠1){displaystyle int {frac {sin ^{n}cx;dx}{cos cx}}=-{frac {sin ^{n-1}cx}{c(n-1)}}+int {frac {sin ^{n-2}cx;dx}{cos cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫sinn⁡cxdxcosm⁡cx=sinn+1⁡cxc(m−1)cosm−1⁡cx−n−m+2m−1∫sinn⁡cxdxcosm−2⁡cx(for m≠1){displaystyle int {frac {sin ^{n}cx;dx}{cos ^{m}cx}}={frac {sin ^{n+1}cx}{c(m-1)cos ^{m-1}cx}}-{frac {n-m+2}{m-1}}int {frac {sin ^{n}cx;dx}{cos ^{m-2}cx}}qquad {mbox{(for }}mneq 1{mbox{)}}}

também: ∫sinn⁡cxdxcosm⁡cx=−sinn−1⁡cxc(n−m)cosm−1⁡cx+n−1n−m∫sinn−2⁡cxdxcosm⁡cx(for m≠n){displaystyle int {frac {sin ^{n}cx;dx}{cos ^{m}cx}}=-{frac {sin ^{n-1}cx}{c(n-m)cos ^{m-1}cx}}+{frac {n-1}{n-m}}int {frac {sin ^{n-2}cx;dx}{cos ^{m}cx}}qquad {mbox{(for }}mneq n{mbox{)}}}

também: ∫sinn⁡cxdxcosm⁡cx=sinn−1⁡cxc(m−1)cosm−1⁡cx−n−1n−1∫sinn−1⁡cxdxcosm−2⁡cx(for m≠1){displaystyle int {frac {sin ^{n}cx;dx}{cos ^{m}cx}}={frac {sin ^{n-1}cx}{c(m-1)cos ^{m-1}cx}}-{frac {n-1}{n-1}}int {frac {sin ^{n-1}cx;dx}{cos ^{m-2}cx}}qquad {mbox{(for }}mneq 1{mbox{)}}}

∫cos⁡cxdxsinn⁡cx=−1c(n−1)sinn−1⁡cx(for n≠1){displaystyle int {frac {cos cx;dx}{sin ^{n}cx}}=-{frac {1}{c(n-1)sin ^{n-1}cx}}qquad {mbox{(for }}nneq 1{mbox{)}}}

∫cos2⁡cxdxsin⁡cx=1c(cos⁡cx+ln⁡|tan⁡cx2|){displaystyle int {frac {cos ^{2}cx;dx}{sin cx}}={frac {1}{c}}left(cos cx+ln left|tan {frac {cx}{2}}right|right)}

∫cos2⁡cxdxsinn⁡cx=−1n−1(cos⁡cxcsinn−1⁡cx)+∫dxsinn−2⁡cx)(for n≠1){displaystyle int {frac {cos ^{2}cx;dx}{sin ^{n}cx}}=-{frac {1}{n-1}}left({frac {cos cx}{csin ^{n-1}cx)}}+int {frac {dx}{sin ^{n-2}cx}}right)qquad {mbox{(for }}nneq 1{mbox{)}}}

∫cosn⁡cxdxsinm⁡cx=−cosn+1⁡cxc(m−1)sinm−1⁡cx−n−m−2m−1∫cosncxdxsinm−2⁡cx(for m≠1){displaystyle int {frac {cos ^{n}cx;dx}{sin ^{m}cx}}=-{frac {cos ^{n+1}cx}{c(m-1)sin ^{m-1}cx}}-{frac {n-m-2}{m-1}}int {frac {cos^{n}cx;dx}{sin ^{m-2}cx}}qquad {mbox{(for }}mneq 1{mbox{)}}}

também: ∫cosn⁡cxdxsinm⁡cx=cosn−1⁡cxc(n−m)sinm−1⁡cx+n−1n−m∫cosn−2cxdxsinm⁡cx(for m≠n){displaystyle int {frac {cos ^{n}cx;dx}{sin ^{m}cx}}={frac {cos ^{n-1}cx}{c(n-m)sin ^{m-1}cx}}+{frac {n-1}{n-m}}int {frac {cos^{n-2}cx;dx}{sin ^{m}cx}}qquad {mbox{(for }}mneq n{mbox{)}}}

também: ∫cosn⁡cxdxsinm⁡cx=−cosn−1⁡cxc(m−1)sinm−1⁡cx−n−1m−1∫cosn−2cxdxsinm−2⁡cx(for m≠1){displaystyle int {frac {cos ^{n}cx;dx}{sin ^{m}cx}}=-{frac {cos ^{n-1}cx}{c(m-1)sin ^{m-1}cx}}-{frac {n-1}{m-1}}int {frac {cos^{n-2}cx;dx}{sin ^{m-2}cx}}qquad {mbox{(for }}mneq 1{mbox{)}}}

Integrais de funções trigonométricas contendo seno e tangente |

∫sen⁡cxtan⁡cxdx=1c(ln⁡|sec⁡cx+tan⁡cx|−sin⁡cx){displaystyle int operatorname {sen} cxtan cx;dx={frac {1}{c}}(ln |sec cx+tan cx|-sin cx)}

∫tann⁡cxdxsin2⁡cx=1c(n−1)tann−1⁡(cx)(for n≠1){displaystyle int {frac {tan ^{n}cx;dx}{sin ^{2}cx}}={frac {1}{c(n-1)}}tan ^{n-1}(cx)qquad {mbox{(for }}nneq 1{mbox{)}}}

Integrais de funções trigonométricas contendo cosseno e tangente |

∫tann⁡cxdxcos2⁡cx=1c(n+1)tann+1⁡cx(for n≠−1){displaystyle int {frac {tan ^{n}cx;dx}{cos ^{2}cx}}={frac {1}{c(n+1)}}tan ^{n+1}cxqquad {mbox{(for }}nneq -1{mbox{)}}}

Integrais de funções trigonométricas contendo seno e cotangente |

∫cotn⁡cxdxsin2⁡cx=1c(n+1)cotn+1⁡cx(for n≠−1){displaystyle int {frac {cot ^{n}cx;dx}{sin ^{2}cx}}={frac {1}{c(n+1)}}cot ^{n+1}cxqquad {mbox{(for }}nneq -1{mbox{)}}}

Integrais de funções trigonométricas contendo cosseno e cotangente |

∫cotn⁡cxdxcos2⁡cx=1c(1−n)tan1−n⁡cx(for n≠1){displaystyle int {frac {cot ^{n}cx;dx}{cos ^{2}cx}}={frac {1}{c(1-n)}}tan ^{1-n}cxqquad {mbox{(for }}nneq 1{mbox{)}}}

Integrais de funções trigonométricas contendo tangente e cotangente |

∫tanm⁡(cx)cotn⁡(cx)dx=1c(m+n−1)tanm+n−1⁡(cx)−∫tanm−2⁡(cx)cotn⁡(cx)dx(for m+n≠1){displaystyle int {frac {tan ^{m}(cx)}{cot ^{n}(cx)}};dx={frac {1}{c(m+n-1)}}tan ^{m+n-1}(cx)-int {frac {tan ^{m-2}(cx)}{cot ^{n}(cx)}};dxqquad {mbox{(for }}m+nneq 1{mbox{)}}}