Definite Integrals of Trigonometric Functions

Basic formulas

\( \int^{\pi/2}_0 \sin^2x\,dx = \int^{\pi/2}_0 \cos^2x\,dx = \frac{\pi}{4} \)

 

\(
\int^\infty_0 \frac{\sin(px)}{x} \,dx = \left\{
\begin{array}{l l l}
\pi/2 & p > 0 \\
~0 & p = 0 \\
-\pi/2 & p < 0 \\
\end{array} \right.
\)

 

\( \int^\infty_0 \frac{\sin^2px}{x^2} = \frac{\pi\,p}{2} \)

 

\( \int^\infty_0 \frac{1 – \cos(px)}{x^2}dx = \frac{\pi\,p}{2} \)

 

\( \int^\infty_0 \frac{\cos(px) – \cos(qx)}{x}dx = ln\frac{q}{p} \)

 

\( \int^\infty_0 \frac{\cos(px) – \cos(qx)}{x^2}dx = \frac{\pi(q-p)}{2} \)

 

\( \int^{2\pi}_0 \frac{dx}{a + b\,\sin x} = \frac{2\pi}{\sqrt{a^2-b^2}} \)

 

\( \int^{2\pi}_0 \frac{dx}{a + b\,\cos(x)} = \frac{2\pi}{\sqrt{a^2-b^2}} \)

 

\( \int^\infty_0 \sin ax^2\,dx = \int^\infty_0 \cos(ax^2)\,dx = \frac{1}{2}\sqrt{\frac{\pi}{2a}} \)

 

\(
\int^\infty_0 \frac{\sin x}{\sqrt{x}} dx =
\int^\infty_0 \frac{\cos x}{\sqrt{x}} dx =
\sqrt{\frac{\pi}{2}}
\)

 

\( \int^\infty_0 \frac{\sin^3x}{x^3} dx = \frac{3\pi}{8} \)

 

\( \int^\infty_0 \frac{\sin^4x}{x^4} dx = \frac{\pi}{3} \)

 

\( \int^\infty_0 \frac{\tan x}{x} dx = \frac{\pi}{2} \)

 

\( \int^{\pi/2}_0 \frac{dx}{a + b\,\cos x} = \frac{\arccos(b/a)}{\sqrt{a^2-b^2}} \)

Advanced formulas

\(
\int^\pi_0 \sin (mx) \cdot \sin (nx)\,dx = \left\{
\begin{array}{l l}
0 & \quad m , n \text{ integers and } m\ne n \\
\pi/2 & \quad m , n \text{ integers and } m = n
\end{array} \right.
\)

 

\(
\int^\pi_0 \cos (mx) \cdot \cos (nx)\,dx = \left\{
\begin{array}{l l}
0 & \quad m , n \text{ integers and } m\ne n \\
\pi/2 & \quad m , n \text{ integers and } m = n
\end{array} \right.
\)

 

\(
\int^\pi_0 \sin (mx) \cdot \cos (nx)\,dx = \left\{
\begin{array}{l l}
0 & \quad m , n \text{ integers and } m + n \text{ odd} \\
2m/(m^2 – n^2) & \quad m , n \text{ integers and } m + n \text{ even}
\end{array} \right.
\)

 

\(
\int^{\pi/2}_0 \sin^{2m}x\,dx = \int^{\pi/2}_0 \cos^{2m}x\,dx =
\frac{1\cdot3\cdot5\dots 2m-1}{2\cdot 4 \cdot 6 \dots 2m} \frac{\pi}{2}
\)

 

\(
\int^{\pi/2}_0 \sin^{2m+1}x\,dx = \int^{\pi/2}_0 \cos^{2m+1}x\,dx =
\frac{2\cdot 4 \cdot 6 \dots 2m}{1 \cdot 3 \cdot 5 \dots 2m + 1}
\)

 

\( \int^{\pi}_0 \sin^{2p-1}x\,\cos^{2q-1}x\,dx = \frac{\Gamma(p)\,\Gamma{q}}{2\,\Gamma(p+q)} \)

 

\(
\int^\infty_0 \frac{\sin (px) \cdot \cos (qx)}{x} \,dx = \left\{
\begin{array}{l l l}
~0 & p > q >0 \\
\pi/2 & 0 < p < q \\
\pi/4 & p = q > 0 \\
\end{array} \right.
\)

 

\(
\int^\infty_0 \frac{\sin (px) \cdot \sin (qx)}{x^2} \,dx = \left\{
\begin{array}{l l l}
\pi\,p/2 & 0 < p \le q \\
\pi\,q/2 & p \ge q > 0 \\
\end{array} \right.
\)

 

\( \int^\infty_0 \frac{\cos (mx)}{x^2 + a^2}dx = \frac{\pi}{2a}e^{-ma} \)

 

\( \int^\infty_0 \frac{x\,\sin (mx)}{x^2 + a^2} dx = \frac{\pi}{2}e^{-ma} \)

 

\( \int^\infty_0 \frac{\sin (mx)}{x\left(x^2 + a^2\right)} dx = \frac{\pi}{2a^2}\left(1-e^{-ma}\right) \)

 

\(
\int^{2\pi}_0 \frac{dx}{(a + b\,\sin x)^2} =
\int^{2\pi}_0 \frac{dx}{(a + b\,\cos x)^2} =
\frac{2\pi\,a}{(a^2 – b^2)^{3/2}}
\)

 

\( \int^{2\pi}_0 \frac{dx}{1 – 2a\,\cos x + a^2} = \frac{2\pi}{1 – a^2}, ~~ 0 < a < 1 \)

 

\(
\int^{\pi}_0 \frac{x\,\sin x\,dx}{1 – 2a\,\cos x + a^2} = \left\{
\begin{array}{l l l}
\frac{\pi}{a} ln(1+a) & |a| < 1 \\
\pi \, ln(1 + \frac{1}{a}) & |a| > 1 \\
\end{array} \right.
\)

 

\( \int^{\pi}_0 \frac{\cos (mx)\,dx}{1 – 2a\,\cos x + a^2} = \frac{\pi a^m}{1 – a^2}, ~~ a^2 < 1 \)

 

\( \int^\infty_0 \sin (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\sin \frac{\pi}{2n} , ~~ n > 1 \)

 

\( \int^\infty_0 \cos (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\cos \frac{\pi}{2n} , ~~ n > 1 \)

 

\( \int^\infty_0 \frac{\sin x}{x^p} dx = \frac{\pi}{2\,\Gamma(p)\, \sin (p\pi/2)}, ~~ 0 < p < 1 \)

 

\( \int^\infty_0 \frac{\cos x}{x^p} dx = \frac{\pi}{2\,\Gamma(p)\, \cos (p\pi/2)}, ~~ 0 < p < 1 \)

 

\(
\int^\infty_0 \sin (ax^2)\,\cos (2bx) \, dx =
\frac{1}{2} \sqrt{\frac{\pi}{2a}} \left(\cos\frac{b^2}{a} – \sin \frac{b^2}{a} \right)
\)

 

\(
\int^\infty_0 \cos (ax^2)\,\cos (2bx) \, dx =
\frac{1}{2} \sqrt{\frac{\pi}{2a}} \left(\cos\frac{b^2}{a} + \sin \frac{b^2}{a} \right)
\)

 

\( \int^\infty_0 \frac{dx}{1 + \tan^mx} dx = \frac{\pi}{4} \)

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