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Math formulas: Definite integrals of trig functions

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Note: In the following formulas all letters are positive.

Basic formulas

0π/2sin2xdx=0π/2cos2xdx=π4 \int^{\pi/2}_0 \sin^2x\,dx = \int^{\pi/2}_0 \cos^2x\,dx = \frac{\pi}{4}
0sin(px)xdx={π/2p>0 0p=0π/2p<0\int^\infty_0 \frac{\sin(px)}{x} \,dx = \left\{ \begin{array}{l l l} \pi/2 & p > 0 \\ ~0 & p = 0 \\ -\pi/2 & p < 0 \\ \end{array} \right.
0sin2pxx2=πp2 \int^\infty_0 \frac{\sin^2px}{x^2} = \frac{\pi\,p}{2}
01cos(px)x2dx=πp2 \int^\infty_0 \frac{1 - \cos(px)}{x^2}dx = \frac{\pi\,p}{2}
0cos(px)cos(qx)xdx=lnqp \int^\infty_0 \frac{\cos(px) - \cos(qx)}{x}dx = ln\frac{q}{p}
0cos(px)cos(qx)x2dx=π(qp)2 \int^\infty_0 \frac{\cos(px) - \cos(qx)}{x^2}dx = \frac{\pi(q-p)}{2}
02πdxa+bsinx=2πa2b2\int^{2\pi}_0 \frac{dx}{a + b\,\sin x} = \frac{2\pi}{\sqrt{a^2-b^2}}
02πdxa+bcos(x)=2πa2b2\int^{2\pi}_0 \frac{dx}{a + b\,\cos(x)} = \frac{2\pi}{\sqrt{a^2-b^2}}
0sinax2dx=0cos(ax2)dx=12π2a\int^\infty_0 \sin ax^2\,dx = \int^\infty_0 \cos(ax^2)\,dx = \frac{1}{2}\sqrt{\frac{\pi}{2a}}
0sinxxdx=0cosxxdx=π2\int^\infty_0 \frac{\sin x}{\sqrt{x}} dx = \int^\infty_0 \frac{\cos x}{\sqrt{x}} dx = \sqrt{\frac{\pi}{2}}
0sin3xx3dx=3π8\int^\infty_0 \frac{\sin^3x}{x^3} dx = \frac{3\pi}{8}
0sin4xx4dx=π3\int^\infty_0 \frac{\sin^4x}{x^4} dx = \frac{\pi}{3}
0tanxxdx=π2\int^\infty_0 \frac{\tan x}{x} dx = \frac{\pi}{2}
0π/2dxa+bcosx=arccos(b/a)a2b2\int^{\pi/2}_0 \frac{dx}{a + b\,\cos x} = \frac{\arccos(b/a)}{\sqrt{a^2-b^2}}

Advanced formulas

0πsin(mx)sin(nx)dx={0m,n integers and mnπ/2m,n integers and m=n\int^\pi_0 \sin (mx) \cdot \sin (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m\ne n \\ \pi/2 & \quad m , n \text{ integers and } m = n \end{array} \right.
0πcos(mx)cos(nx)dx={0m,n integers and mnπ/2m,n integers and m=n\int^\pi_0 \cos (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m\ne n \\ \pi/2 & \quad m , n \text{ integers and } m = n \end{array} \right.
0πsin(mx)cos(nx)dx={0m,n integers and m+n odd2m/(m2n2)m,n integers and m+n even\int^\pi_0 \sin (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m + n \text{ odd} \\ 2m/(m^2 - n^2) & \quad m , n \text{ integers and } m + n \text{ even} \end{array} \right.
0π/2sin2mxdx=0π/2cos2mxdx=1352m12462mπ2 \int^{\pi/2}_0 \sin^{2m}x\,dx = \int^{\pi/2}_0 \cos^{2m}x\,dx = \frac{1\cdot3\cdot5\dots 2m-1}{2\cdot 4 \cdot 6 \dots 2m} \frac{\pi}{2}
0π/2sin2m+1xdx=0π/2cos2m+1xdx=2462m1352m+1 \int^{\pi/2}_0 \sin^{2m+1}x\,dx = \int^{\pi/2}_0 \cos^{2m+1}x\,dx = \frac{2\cdot 4 \cdot 6 \dots 2m}{1 \cdot 3 \cdot 5 \dots 2m + 1}
0πsin2p1xcos2q1xdx=Γ(p)Γq2Γ(p+q) \int^{\pi}_0 \sin^{2p-1}x\,\cos^{2q-1}x\,dx = \frac{\Gamma(p)\,\Gamma{q}}{2\,\Gamma(p+q)}
0sin(px)cos(qx)xdx={ 0p>q>0π/20<p<qπ/4p=q>0\int^\infty_0 \frac{\sin (px) \cdot \cos (qx)}{x} \,dx = \left\{ \begin{array}{l l l} ~0 & p > q >0 \\ \pi/2 & 0 < p < q \\ \pi/4 & p = q > 0 \\ \end{array} \right.
0sin(px)sin(qx)x2dx={πp/20<pqπq/2pq>0\int^\infty_0 \frac{\sin (px) \cdot \sin (qx)}{x^2} \,dx = \left\{ \begin{array}{l l l} \pi\,p/2 & 0 < p \le q \\ \pi\,q/2 & p \ge q > 0 \\ \end{array} \right.
0cos(mx)x2+a2dx=π2aema \int^\infty_0 \frac{\cos (mx)}{x^2 + a^2}dx = \frac{\pi}{2a}e^{-ma}
0xsin(mx)x2+a2dx=π2ema \int^\infty_0 \frac{x\,\sin (mx)}{x^2 + a^2} dx = \frac{\pi}{2}e^{-ma}
0sin(mx)x(x2+a2)dx=π2a2(1ema)\int^\infty_0 \frac{\sin (mx)}{x\left(x^2 + a^2\right)} dx = \frac{\pi}{2a^2}\left(1-e^{-ma}\right)
02πdx(a+bsinx)2=02πdx(a+bcosx)2=2πa(a2b2)3/2\int^{2\pi}_0 \frac{dx}{(a + b\,\sin x)^2} = \int^{2\pi}_0 \frac{dx}{(a + b\,\cos x)^2} = \frac{2\pi\,a}{(a^2 - b^2)^{3/2}}
02πdx12acosx+a2=2π1a2,  0<a<1\int^{2\pi}_0 \frac{dx}{1 - 2a\,\cos x + a^2} = \frac{2\pi}{1 - a^2}, ~~ 0 < a < 1
0πxsinxdx12acosx+a2={πaln(1+a)a<1πln(1+1a)a>1\int^{\pi}_0 \frac{x\,\sin x\,dx}{1 - 2a\,\cos x + a^2} = \left\{ \begin{array}{l l l} \frac{\pi}{a} ln(1+a) & |a| < 1 \\ \pi \, ln(1 + \frac{1}{a}) & |a| > 1 \\ \end{array} \right.
0πcos(mx)dx12acosx+a2=πam1a2,  a2<1\int^{\pi}_0 \frac{\cos (mx)\,dx}{1 - 2a\,\cos x + a^2} = \frac{\pi a^m}{1 - a^2}, ~~ a^2 < 1
0sin(axn)dx=1na1/nΓ(1/n)sinπ2n,  n>1\int^\infty_0 \sin (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\sin \frac{\pi}{2n} , ~~ n > 1
0cos(axn)dx=1na1/nΓ(1/n)cosπ2n,  n>1\int^\infty_0 \cos (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\cos \frac{\pi}{2n} , ~~ n > 1
0sinxxpdx=π2Γ(p)sin(pπ/2),  0<p<1\int^\infty_0 \frac{\sin x}{x^p} dx = \frac{\pi}{2\,\Gamma(p)\, \sin (p\pi/2)}, ~~ 0 < p < 1
0cosxxpdx=π2Γ(p)cos(pπ/2),  0<p<1\int^\infty_0 \frac{\cos x}{x^p} dx = \frac{\pi}{2\,\Gamma(p)\, \cos (p\pi/2)}, ~~ 0 < p < 1
0sin(ax2)cos(2bx)dx=12π2a(cosb2asinb2a)\int^\infty_0 \sin (ax^2)\,\cos (2bx) \, dx = \frac{1}{2} \sqrt{\frac{\pi}{2a}} \left(\cos\frac{b^2}{a} - \sin \frac{b^2}{a} \right)
0cos(ax2)cos(2bx)dx=12π2a(cosb2a+sinb2a)\int^\infty_0 \cos (ax^2)\,\cos (2bx) \, dx = \frac{1}{2} \sqrt{\frac{\pi}{2a}} \left(\cos\frac{b^2}{a} + \sin \frac{b^2}{a} \right)
0dx1+tanmxdx=π4\int^\infty_0 \frac{dx}{1 + \tan^mx} dx = \frac{\pi}{4}

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