Calculus Tables
Derivative and integral tables.
This page contains tables of derivatives and integrals most commonly encountered in Calculus I - III. More specialized tables may be found in other pages.
d d x c = 0 \begin{align}
\dv{}{x}c = 0
\end{align} d x d c = 0 d d x [ f ( x ) + g ( x ) ] = f ′ ( x ) + g ′ ( x ) \begin{align}
\dv{}{x}\left[f(x)+g(x)\right] = f'(x) + g'(x)
\end{align} d x d [ f ( x ) + g ( x ) ] = f ′ ( x ) + g ′ ( x ) d d x [ f ( x ) g ( x ) ] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) \begin{align}
\dv{}{x}\left[f(x)g(x)\right] = f'(x)g(x) + f(x)g'(x)
\end{align} d x d [ f ( x ) g ( x ) ] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) d d x f ( x ) g ( x ) = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) [ g ( x ) ] 2 \begin{align}
\dv{}{x}\frac{f(x)}{g(x)} = \frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}
\end{align} d x d g ( x ) f ( x ) = [ g ( x ) ] 2 g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) d d x f ( g ( x ) ) = f ′ ( g ( x ) ) g ′ ( x ) \begin{align}
\dv{}{x}f(g(x)) = f'(g(x))g'(x)
\end{align} d x d f ( g ( x )) = f ′ ( g ( x )) g ′ ( x ) d d x x n = n x n − 1 \begin{align}
\dv{}{x}x^n = nx^{n-1}
\end{align} d x d x n = n x n − 1 Exponential and Logarithmic Functions
d d x e x = e x \begin{align}
\dv{}{x} e^x = e^x
\end{align} d x d e x = e x d d x a x = a x ln a , a > 0 \begin{align}
\dv{}{x} a^x = a^x \ln a,\quad a > 0
\end{align} d x d a x = a x ln a , a > 0 d d x ln ∣ x ∣ = 1 x , ∣ x ∣ ≠ 0 \begin{align}
\dv{}{x} \ln |x| = \frac{1}{x},\quad |x| \neq 0
\end{align} d x d ln ∣ x ∣ = x 1 , ∣ x ∣ = 0 d d x log a x = 1 x ln a , a > 0 \begin{align}
\dv{}{x} \log_a x = \frac{1}{x \ln a},\quad a > 0
\end{align} d x d log a x = x ln a 1 , a > 0 Trigonometric Functions
d d x sin x = cos x \begin{align}
\dv{}{x} \sin x = \cos x
\end{align} d x d sin x = cos x d d x cos x = − sin x \begin{align}
\dv{}{x} \cos x = -\sin x
\end{align} d x d cos x = − sin x d d x tan x = sec 2 x = 1 + tan 2 x \begin{align}
\dv{}{x} \tan x = \sec^2 x = 1+\tan^2 x
\end{align} d x d tan x = sec 2 x = 1 + tan 2 x d d x csc x = − csc x cot x \begin{align}
\dv{}{x} \csc x = -\csc x \cot x
\end{align} d x d csc x = − csc x cot x d d x sec x = sec x tan x \begin{align}
\dv{}{x} \sec x = \sec x \tan x
\end{align} d x d sec x = sec x tan x d d x cot x = − csc 2 x = − 1 − cot 2 x \begin{align}
\dv{}{x} \cot x = -\csc^2 x = -1-\cot^2 x
\end{align} d x d cot x = − csc 2 x = − 1 − cot 2 x Inverse Trigonometric Functions
d d x sin − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}}
\end{align} d x d sin − 1 x = 1 − x 2 1 d d x cos − 1 x = − 1 1 − x 2 \begin{align}
\dv{}{x} \cos^{-1} x = -\frac{1}{\sqrt{1-x^2}}
\end{align} d x d cos − 1 x = − 1 − x 2 1 d d x tan − 1 x = 1 1 + x 2 \begin{align}
\dv{}{x} \tan^{-1} x = \frac{1}{1+x^2}
\end{align} d x d tan − 1 x = 1 + x 2 1 d d x csc − 1 x = − 1 ∣ x ∣ x 2 − 1 \begin{align}
\dv{}{x} \csc^{-1} x = -\frac{1}{|x|\sqrt{x^2-1}}
\end{align} d x d csc − 1 x = − ∣ x ∣ x 2 − 1 1 d d x sec − 1 x = 1 ∣ x ∣ x 2 − 1 \begin{align}
\dv{}{x} \sec^{-1} x = \frac{1}{|x|\sqrt{x^2-1}}
\end{align} d x d sec − 1 x = ∣ x ∣ x 2 − 1 1 d d x cot − 1 x = − 1 1 + x 2 \begin{align}
\dv{}{x} \cot^{-1} x = -\frac{1}{1+x^2}
\end{align} d x d cot − 1 x = − 1 + x 2 1 Hyperbolic Functions
d d x sinh x = cosh x \begin{align}
\dv{}{x} \sinh x = \cosh x
\end{align} d x d sinh x = cosh x d d x cosh x = sinh x \begin{align}
\dv{}{x} \cosh x = \sinh x
\end{align} d x d cosh x = sinh x d d x tanh x = s e c h 2 x = 1 − tanh 2 x \begin{align}
\dv{}{x} \tanh x = \sech^2\,x = 1-\tanh^2 x
\end{align} d x d tanh x = sech 2 x = 1 − tanh 2 x d d x c s c h x = − c s c h x coth x \begin{align}
\dv{}{x} \csch\,x = -\csch\,x \coth x
\end{align} d x d csch x = − csch x coth x d d x s e c h x = − s e c h x tanh x \begin{align}
\dv{}{x} \sech\,x = -\sech\,x \tanh x
\end{align} d x d sech x = − sech x tanh x d d x coth x = − c s c h 2 x = 1 − coth 2 x \begin{align}
\dv{}{x} \coth x = -\csch^2\,x = 1-\coth^2 x
\end{align} d x d coth x = − csch 2 x = 1 − coth 2 x Inverse Hyperbolic Functions
d d x sinh − 1 x = 1 1 + x 2 \begin{align}
\dv{}{x} \sinh^{-1} x = \frac{1}{\sqrt{1+x^2}}
\end{align} d x d sinh − 1 x = 1 + x 2 1 d d x cosh − 1 x = 1 x 2 − 1 \begin{align}
\dv{}{x} \cosh^{-1} x = \frac{1}{\sqrt{x^2-1}}
\end{align} d x d cosh − 1 x = x 2 − 1 1 d d x tanh − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \tanh^{-1} x = \frac{1}{1-x^2}
\end{align} d x d tanh − 1 x = 1 − x 2 1 d d x c s c h − 1 x = − 1 ∣ x ∣ 1 + x 2 \begin{align}
\dv{}{x} \csch^{-1}\,x = -\frac{1}{|x|\sqrt{1+x^2}}
\end{align} d x d csch − 1 x = − ∣ x ∣ 1 + x 2 1 d d x s e c h − 1 x = − 1 x 1 − x 2 \begin{align}
\dv{}{x} \sech^{-1}\,x = -\frac{1}{x\sqrt{1-x^2}}
\end{align} d x d sech − 1 x = − x 1 − x 2 1 d d x coth − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \coth^{-1} x = \frac{1}{1-x^2}
\end{align} d x d coth − 1 x = 1 − x 2 1 Higher-Order Derivatives
General Leibniz rule: If f ( x ) f(x) f ( x ) g ( x ) g(x) g ( x ) n n n
d n d x n [ f ( x ) g ( x ) ] = ∑ k = 0 n ( n k ) d n − k d x n − k f ( x ) d k d x k g ( x ) . \begin{align}
\ndv{n}{}{x}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} \ndv{n-k}{}{x}f(x)\ndv{k}{}{x}g(x).
\end{align} d x n d n [ f ( x ) g ( x )] = k = 0 ∑ n ( k n ) d x n − k d n − k f ( x ) d x k d k g ( x ) . ∫ x n d x = x n + 1 n + 1 + C n ≠ − 1 \begin{align}
\int x^n\,\dd{x} = \frac{x^{n+1}}{n+1}+C\quad n \neq -1
\end{align} ∫ x n d x = n + 1 x n + 1 + C n = − 1 ∫ 1 x d x = ln ∣ x ∣ + C \begin{align}
\int \frac{1}{x}\,\dd{x} = \ln|x| + C
\end{align} ∫ x 1 d x = ln ∣ x ∣ + C ∫ 1 x 2 + a 2 d x = 1 a tan − 1 x a + C \begin{align}
\int \frac{1}{x^2+a^2}\,\dd{x} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C
\end{align} ∫ x 2 + a 2 1 d x = a 1 tan − 1 a x + C ∫ 1 x 2 − a 2 d x = 1 2 a ln ∣ x − a x + a ∣ + C = { − 1 a tanh − 1 x a + C = 1 2 a ln a − x a + x + C , ∣ x ∣ < ∣ a ∣ − 1 a coth − 1 x a + C = 1 2 a ln x − a x + a + C , ∣ x ∣ > ∣ a ∣ \begin{align}
\int \frac{1}{x^2-a^2}\,\dd{x} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C = \begin{cases}
-\dfrac{1}{a}\tanh^{-1}\dfrac{x}{a} + C = \dfrac{1}{2a}\ln\dfrac{a-x}{a+x} + C,\quad |x|<|a| \\[1em]
-\dfrac{1}{a}\coth^{-1}\dfrac{x}{a} + C = \dfrac{1}{2a}\ln\dfrac{x-a}{x+a} + C,\quad |x|>|a|
\end{cases}
\end{align} ∫ x 2 − a 2 1 d x = 2 a 1 ln x + a x − a + C = ⎩ ⎨ ⎧ − a 1 tanh − 1 a x + C = 2 a 1 ln a + x a − x + C , ∣ x ∣ < ∣ a ∣ − a 1 coth − 1 a x + C = 2 a 1 ln x + a x − a + C , ∣ x ∣ > ∣ a ∣ ∫ x x 2 + a 2 d x = 1 2 ln ∣ x 2 + a 2 ∣ + C \begin{align}
\int \frac{x}{x^2+a^2}\,\dd{x} = \frac{1}{2}\ln\left|x^2+a^2\right| + C
\end{align} ∫ x 2 + a 2 x d x = 2 1 ln x 2 + a 2 + C ∫ x 2 x 2 + a 2 d x = x − a tan − 1 x a + C \begin{align}
\int \frac{x^2}{x^2+a^2}\,\dd{x} = x - a\tan^{-1}\frac{x}{a} + C
\end{align} ∫ x 2 + a 2 x 2 d x = x − a tan − 1 a x + C Exponential and Logarithmic Functions
∫ e a x d x = 1 a e a x + C \begin{align}
\int e^{ax}\,\dd{x} = \frac{1}{a}e^{ax} + C
\end{align} ∫ e a x d x = a 1 e a x + C ∫ a x d x = a x ln a + C , a > 0 \begin{align}
\int a^{x}\,\dd{x} = \frac{a^x}{\ln a} + C,\quad a > 0
\end{align} ∫ a x d x = ln a a x + C , a > 0 ∫ ln x d x = x ln x − x + C \begin{align}
\int \ln x\,\dd{x} = x\ln x - x + C
\end{align} ∫ ln x d x = x ln x − x + C ∫ x n ln x d x = x n + 1 ( n + 1 ) 2 [ ( n + 1 ) ln x − 1 ] + C , n ≠ − 1 \begin{align}
\int x^n \ln x\,\dd{x} = \frac{x^{n+1}}{(n+1)^2}\left[(n+1)\ln x-1\right] + C,\quad n \neq -1
\end{align} ∫ x n ln x d x = ( n + 1 ) 2 x n + 1 [ ( n + 1 ) ln x − 1 ] + C , n = − 1 ∫ log a x d x = x ln a ( ln x − 1 ) + C , a > 0 \begin{align}
\int \log_a x\,\dd{x} = \frac{x}{\ln a}(\ln x - 1) + C,\quad a > 0
\end{align} ∫ log a x d x = ln a x ( ln x − 1 ) + C , a > 0 Trigonometric Functions
∫ sin x d x = − cos x + C \begin{align}
\int \sin x\,\dd{x} = -\cos x + C
\end{align} ∫ sin x d x = − cos x + C ∫ cos x d x = sin x + C \begin{align}
\int \cos x\,\dd{x} = \sin x + C
\end{align} ∫ cos x d x = sin x + C ∫ tan x d x = − ln ∣ cos x ∣ + C = ln ∣ sec x ∣ + C \begin{align}
\int \tan x\,\dd{x} = -\ln|\cos x| + C = \ln|\sec x| + C
\end{align} ∫ tan x d x = − ln ∣ cos x ∣ + C = ln ∣ sec x ∣ + C ∫ csc x d x = − ln ∣ csc x + cot x ∣ + C = ln ∣ csc x − cot x ∣ + C = ln ∣ tan x 2 ∣ + C \begin{align}
\int \csc x\,\dd{x} = -\ln|\csc x+\cot x| + C = \ln|\csc x-\cot x| + C = \ln\left|\tan\frac{x}{2}\right| + C
\end{align} ∫ csc x d x = − ln ∣ csc x + cot x ∣ + C = ln ∣ csc x − cot x ∣ + C = ln tan 2 x + C ∫ sec x d x = ln ∣ sec x + tan x ∣ + C = ln ∣ tan ( x 2 + π 4 ) ∣ + C \begin{align}
\int \sec x\,\dd{x} = \ln|\sec x+\tan x| + C = \ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\right| + C
\end{align} ∫ sec x d x = ln ∣ sec x + tan x ∣ + C = ln tan ( 2 x + 4 π ) + C ∫ cot x d x = ln ∣ sin x ∣ + C = − ln ∣ csc x ∣ + C \begin{align}
\int \cot x\,\dd{x} = \ln|\sin x| + C = -\ln|\csc x| + C
\end{align} ∫ cot x d x = ln ∣ sin x ∣ + C = − ln ∣ csc x ∣ + C ∫ sin 2 x d x = 1 2 ( x − sin 2 x 2 ) + C \begin{align}
\int \sin^2 x\,\dd{x} = \frac{1}{2}\left(x-\frac{\sin 2x}{2}\right) + C
\end{align} ∫ sin 2 x d x = 2 1 ( x − 2 sin 2 x ) + C ∫ cos 2 x d x = 1 2 ( x + sin 2 x 2 ) + C \begin{align}
\int \cos^2 x\,\dd{x} = \frac{1}{2}\left(x+\frac{\sin 2x}{2}\right) + C
\end{align} ∫ cos 2 x d x = 2 1 ( x + 2 sin 2 x ) + C ∫ tan 2 x d x = tan x − x + C \begin{align}
\int \tan^2 x\,\dd{x} = \tan x - x + C
\end{align} ∫ tan 2 x d x = tan x − x + C ∫ csc 2 x = − cot x + C \begin{align}
\int \csc^2 x = -\cot x + C
\end{align} ∫ csc 2 x = − cot x + C ∫ sec 2 x = tan x + C \begin{align}
\int \sec^2 x = \tan x + C
\end{align} ∫ sec 2 x = tan x + C ∫ cot 2 x = − cot x − x + C \begin{align}
\int \cot^2 x = -\cot x - x + C
\end{align} ∫ cot 2 x = − cot x − x + C ∫ sin n x d x = − s i n n − 1 x cos x n + n − 1 n ∫ sin n − 2 x d x \begin{align}
\int \sin^n x\,\dd{x} = -\frac{sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int \sin^{n-2} x\,\dd{x}
\end{align} ∫ sin n x d x = − n s i n n − 1 x cos x + n n − 1 ∫ sin n − 2 x d x ∫ cos n x d x = cos n − 1 x sin x n + n − 1 n ∫ cos n − 2 x d x \begin{align}
\int \cos^n x\,\dd{x} = \frac{\cos^{n-1}x\sin x}{n}+\frac{n-1}{n}\int \cos^{n-2} x\,\dd{x}
\end{align} ∫ cos n x d x = n cos n − 1 x sin x + n n − 1 ∫ cos n − 2 x d x Inverse Trigonometric Functions
∫ arcsin x d x = x arcsin x + 1 − x 2 + C , ∣ x ∣ ≤ 1 \begin{align}
\int \arcsin x\,\dd{x} = x\arcsin x+\sqrt{1-x^2} + C,\quad |x| \leq 1
\end{align} ∫ arcsin x d x = x arcsin x + 1 − x 2 + C , ∣ x ∣ ≤ 1 ∫ arccos x d x = x arccos x − 1 − x 2 + C , ∣ x ∣ ≤ 1 \begin{align}
\int \arccos x\,\dd{x} = x\arccos x-\sqrt{1-x^2} + C,\quad |x| \leq 1
\end{align} ∫ arccos x d x = x arccos x − 1 − x 2 + C , ∣ x ∣ ≤ 1 ∫ arctan x d x = x arctan x − 1 2 ln ∣ 1 + x 2 ∣ + C \begin{align}
\int \arctan x\,\dd{x} = x\arctan x-\frac{1}{2}\ln\left|1+x^2\right| + C
\end{align} ∫ arctan x d x = x arctan x − 2 1 ln 1 + x 2 + C ∫ a r c c s c x d x = x a r c c s c x + 1 2 ln ∣ x ( 1 + 1 − x − 2 ) ∣ + C , ∣ x ∣ ≥ 1 \begin{align}
\int \arccsc\,x\,\dd{x} = x\,\arccsc\,x+\frac{1}{2}\ln\left|x\left(1+\sqrt{1-x^{-2}}\right)\right| + C,\quad |x| \geq 1
\end{align} ∫ arccsc x d x = x arccsc x + 2 1 ln x ( 1 + 1 − x − 2 ) + C , ∣ x ∣ ≥ 1 ∫ a r c s e c x d x = x a r c s e c x − 1 2 ln ∣ x ( 1 + 1 − x − 2 ) ∣ + C , ∣ x ∣ ≥ 1 \begin{align}
\int \arcsec\,x\,\dd{x} = x\,\arcsec\,x-\frac{1}{2}\ln\left|x\left(1+\sqrt{1-x^{-2}}\right)\right| + C,\quad |x| \geq 1
\end{align} ∫ arcsec x d x = x arcsec x − 2 1 ln x ( 1 + 1 − x − 2 ) + C , ∣ x ∣ ≥ 1 ∫ a r c c o t x d x = x a r c c o t x + 1 2 ln ∣ 1 + x 2 ∣ + C \begin{align}
\int \arccot x\,\dd{x} = x\,\arccot\,x+\frac{1}{2}\ln\left|1+x^2\right| + C
\end{align} ∫ arccot x d x = x arccot x + 2 1 ln 1 + x 2 + C Hyperbolic Functions
∫ sinh x d x = cosh x + C \begin{align}
\int \sinh x\,\dd{x} = \cosh x + C
\end{align} ∫ sinh x d x = cosh x + C ∫ cosh x d x = sinh x + C \begin{align}
\int \cosh x\,\dd{x} = \sinh x + C
\end{align} ∫ cosh x d x = sinh x + C ∫ tanh x d x = ln ( cosh x ) + C \begin{align}
\int \tanh x\,\dd{x} = \ln(\cosh x) + C
\end{align} ∫ tanh x d x = ln ( cosh x ) + C ∫ c s c h x d x = ln ∣ coth x − c s c h x ∣ + C = ln ∣ tanh x 2 ∣ + C , x ≠ 0 \begin{align}
\int \csch\,x\,\dd{x} = \ln|\coth x-\csch\,x| + C = \ln\left|\tanh\frac{x}{2}\right| + C,\quad x \neq 0
\end{align} ∫ csch x d x = ln ∣ coth x − csch x ∣ + C = ln tanh 2 x + C , x = 0 ∫ s e c h x d x = arctan ( sinh x ) + C \begin{align}
\int \sech\,x\,\dd{x} = \arctan(\sinh x) + C
\end{align} ∫ sech x d x = arctan ( sinh x ) + C ∫ coth x d x = ln ∣ sinh x ∣ + C , x ≠ 0 \begin{align}
\int \coth x\,\dd{x} = \ln|\sinh x| + C,\quad x \neq 0
\end{align} ∫ coth x d x = ln ∣ sinh x ∣ + C , x = 0 ∫ c s c h 2 x d x = − coth x + C \begin{align}
\int \csch^2\,x\,\dd{x} = -\coth x + C
\end{align} ∫ csch 2 x d x = − coth x + C ∫ s e c h 2 x d x = tanh x + C \begin{align}
\int \sech^2\,x\,\dd{x} = \tanh x + C
\end{align} ∫ sech 2 x d x = tanh x + C Inverse Hyperbolic Functions
∫ a r c s i n h x d x = x a r c s i n h x − 1 + x 2 + C \begin{align}
\int \arcsinh\,x\,\dd{x} = x\,\arcsinh\,x-\sqrt{1+x^2} + C
\end{align} ∫ arcsinh x d x = x arcsinh x − 1 + x 2 + C ∫ a r c c o s h x d x = x a r c c o s h x − x 2 − 1 + C , x ≥ 1 \begin{align}
\int \arccosh\,x\,\dd{x} = x\,\arccosh\,x-\sqrt{x^2-1} + C,\quad x \geq 1
\end{align} ∫ arccosh x d x = x arccosh x − x 2 − 1 + C , x ≥ 1 ∫ a r c t a n h x d x = x a r c t a n h x + 1 2 ln ( 1 − x 2 ) + C , ∣ x ∣ < 1 \begin{align}
\int \arctanh\,x\,\dd{x} = x\,\arctanh\,x+\frac{1}{2}\ln\left(1-x^2\right) + C,\quad |x| < 1
\end{align} ∫ arctanh x d x = x arctanh x + 2 1 ln ( 1 − x 2 ) + C , ∣ x ∣ < 1 ∫ a r c c s c h x d x = x a r c c s c h x + ∣ a r c s i n h x ∣ + C , x ≠ 0 \begin{align}
\int \arccsch\,x\,\dd{x} = x\,\arccsch\,x+\left|\arcsinh\,x\right| + C,\quad x \neq 0
\end{align} ∫ arccsch x d x = x arccsch x + ∣ arcsinh x ∣ + C , x = 0 ∫ a r c s e c h x d x = x a r c s e c h x + arcsin x + C , 0 < x ≤ 1 \begin{align}
\int \arcsech\,x\,\dd{x} = x\,\arcsech\,x+\arcsin x + C,\quad 0 < x \leq 1
\end{align} ∫ arcsech x d x = x arcsech x + arcsin x + C , 0 < x ≤ 1 ∫ a r c c o t h x d x = x a r c c o t h x + 1 2 ln ( x 2 − 1 ) + C , ∣ x ∣ > 1 \begin{align}
\int \arccoth\,x\,\dd{x} = x\,\arccoth\,x+\frac{1}{2}\ln\left(x^2-1\right) + C,\quad |x| > 1
\end{align} ∫ arccoth x d x = x arccoth x + 2 1 ln ( x 2 − 1 ) + C , ∣ x ∣ > 1 Functions Involving Radicals
∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 ln ( x + a 2 + x 2 ) + C \begin{align}
\int \sqrt{a^2+x^2}\,\dd{x} = \frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\ln\left(x+\sqrt{a^2+x^2}\right) + C
\end{align} ∫ a 2 + x 2 d x = 2 x a 2 + x 2 + 2 a 2 ln ( x + a 2 + x 2 ) + C ∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 ln ∣ x + x 2 − a 2 ∣ + C \begin{align}
\int \sqrt{x^2-a^2}\,\dd{x} = \frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2-a^2}\right| + C
\end{align} ∫ x 2 − a 2 d x = 2 x x 2 − a 2 − 2 a 2 ln x + x 2 − a 2 + C ∫ 1 a 2 + x 2 d x = ln ( x + a 2 + x 2 ) + C \begin{align}
\int \frac{1}{\sqrt{a^2+x^2}}\,\dd{x} = \ln\left(x+\sqrt{a^2+x^2}\right) + C
\end{align} ∫ a 2 + x 2 1 d x = ln ( x + a 2 + x 2 ) + C ∫ 1 a 2 − x 2 d x = sin − 1 x a + C \begin{align}
\int \frac{1}{\sqrt{a^2-x^2}}\,\dd{x} = \sin^{-1}\frac{x}{a} + C
\end{align} ∫ a 2 − x 2 1 d x = sin − 1 a x + C ∫ 1 x 2 − a 2 d x = ln ∣ x + x 2 − a 2 ∣ + C \begin{align}
\int \frac{1}{\sqrt{x^2-a^2}}\,\dd{x} = \ln\left|x+\sqrt{x^2-a^2}\right| + C
\end{align} ∫ x 2 − a 2 1 d x = ln x + x 2 − a 2 + C ∫ 1 x x 2 − a 2 d x = 1 a sec − 1 x a + C \begin{align}
\int \frac{1}{x\sqrt{x^2-a^2}}\,\dd{x} = \frac{1}{a}\sec^{-1}\frac{x}{a} + C
\end{align} ∫ x x 2 − a 2 1 d x = a 1 sec − 1 a x + C ∫ 1 x a 2 − x 2 d x = − 1 a ln ∣ a + a 2 − x 2 x ∣ + C \begin{align}
\int \frac{1}{x\sqrt{a^2-x^2}}\,\dd{x} = -\frac{1}{a}\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right| + C
\end{align} ∫ x a 2 − x 2 1 d x = − a 1 ln x a + a 2 − x 2 + C