Usual functions
| \(f(x)\) | \(F(x)\) |
| $$\lambda \text{ (constant)}$$ | $$\lambda .x + C$$ |
| $$ x $$ | $$\frac{x^2}{2}+C$$ |
| $$ x^n $$ | $$\frac{x^{n+1}}{n+1}+ C$$ |
| $$ \frac{1}{x}=x^{-1} $$ | $$ln|x|$$ |
| $$ \frac{1}{x^n}=x^{-n} $$ | $$\frac{-1}{(n-1).x^{n-1}} + C$$ |
| $$ \frac{1}{\sqrt{x}}$$ | $$2 \sqrt{x} + C$$ |
| $$ ln(x) $$ | $$x.ln(x)-x + C$$ |
| $$ e^x $$ | $$e^x + C$$ |
| $$sin(x)$$ | $$- cos(x)+ C$$ |
| $$cos(x)$$ | $$sin(x )+ C$$ |
| $$ 1+ tan^2(x) = \frac{1}{cos^2(x)}$$ | $$tan(x)$$ |
| $$sin(a.x)$$ | $$-\frac{1}{a}.cos(a.x) + C$$ |
| $$cos(a.x)$$ | $$\frac{1}{a}.sin(a.x) + C$$ |
| $$tan(x)$$ | $$-ln(|cos(x)|)+ C$$ |
| $$ e^{a.x} $$ | $$\frac{1}{a}.e^{a.x}+ C$$ |
| $$a^x$$ | $$\frac{1}{ln(a)}.a^{x}+ C$$ |
| $$ \frac{1}{x^2+1}$$ | $$2 arctan x + C$$ |
| $$ (x-a)^n $$ | $$\frac{1}{n+1}.(x-a)^{n+1} + C$$ |
| $$ \frac{1}{x-a} $$ | $$ln(|x-a|)+ C$$ |
| $$ \frac{1}{(x-a)^n} $$ | $$\frac{-1}{(n-1)(x-a)^{n+1}}+ C$$ |
| $$ \sqrt{x-a}$$ | $$\frac{2}{3}.(x-a)^{\frac{3}{2}}+ C$$ |
| $$\frac{1}{\sqrt{x-a}}$$ | $$2.\sqrt{x-a}+ C$$ |
| $$ \frac{1}{\sqrt{1-x^2}} $$ | $$ arcsin(x) + C$$ |
| $$ \frac{-1}{\sqrt{1-x^2}} $$ | $$ arccos(x) + C$$ |
| $$ sinh(x) $$ | $$cosh(x)+ C$$ |
| $$ cosh(x) $$ | $$ sinh(x)+ C$$ |
| $$ 1-tanh^2(x)= \frac{1}{cosh^2(x)} $$ | $$ tanh(x)+ C$$ |
Operations
| \(f(x)\) | \(F(x)\) |
| $$ u’.u^n$$ | $$ \frac{u^{n+1}}{n+1} +C$$ |
| $$ \frac{u’}{u^2}$$ | $$ -\frac{1}{u} +C$$ |
| $$ \frac{u’}{u^n}$$ | $$ -\frac{1}{(n-1).u^{n-1}} +C (n \in \mathbb{N}, n \geq 2)$$ |
| $$ \frac{u’}{\sqrt{u}}$$ | $$ 2.\sqrt{u} +C$$ |
| $$ \frac{u’}{u}$$ | $$ ln|u| +C$$ |
| $$ -u’.sin(u)$$ | $$ cos(u) +C$$ |
| $$u’.cos(u)$$ | $$ sin(u) +C$$ |
| $$ u’.e^u$$ | $$ e^u +C$$ |
| $$ u’.u^a$$ | $$ \frac{1}{a+1}.u^{a+1}+C if a \in \mathbb{R}, \text{-1 excluded}$$ |
| $$ u’.u^a$$ | $$ ln(u)+C\text{ if }a=-1$$ |
| $$ u’.(1+tan^2(u))=\frac{u’}{cos^2(u)}$$ | $$ e^u +C$$ |
| $$ \frac{u’}{\sqrt{1-u^2}}$$ | $$ arcsin(u) +C$$ |
| $$ \frac{-u’}{\sqrt{1-u^2}}$$ | $$ arccos(u) +C$$ |
| $$ \frac{u’}{u^2+1}$$ | $$ arctan(u) +C$$ |
| $$u’.cosh(u)$$ | $$ sinh(u) +C$$ |
| $$u’.sinh(u)$$ | $$ cosh(u) +C$$ |
| $$ u’.(1-tanh^2(u))=\frac{u’}{cosh^2(u)}$$ | $$ tanh(u) +C$$ |
Trigonometric functions
| Primitive \(F(x)\) | Function \(f(x)\) | Derivative \(f'(x)\) |
| $$-cos(x)$$ | $$sin(x)$$ | $$cos(x)$$ |
| $$sin(x)$$ | $$cos(x)$$ | $$-sin(x)$$ |
| $$ln(|sec(x)|)$$ | $$tan(x)$$ | $$sec^2(x)$$ |
| $$ln(|sin(x)|)$$ | $$cotan(x)$$ | $$-cosec^2(x)$$ |
| $$\frac{1}{2}.ln(\frac{|1+sin(x)|}{|1-sin(x)|})$$ | $$sec(x)$$ | $$sec(x).tan(x)$$ |
| $$\frac{1}{2}.ln(\frac{|1-cos(x)|}{|1+cos(x)|})$$ | $$cosec(x)$$ | $$-cosec(x).cotan(x)$$ |
| $$x.arcsin(x)+\sqrt{1-x^2}$$ | $$arcsin(x)$$ | $$\frac{1}{\sqrt{1-x^2}}$$ |
| $$x.arccos(x)-\sqrt{1-x^2}$$ | $$arccos(x)$$ | $$\frac{-1}{\sqrt{1-x^2}}$$ |
| $$x.arctan(x)-\frac{1}{2}.ln(1+x^2)$$ | $$arctan(x)$$ | $$\frac{1}{x^2+1}$$ |
| $$x.arccotan(x)+\frac{1}{2}.ln(1+x^2)$$ | $$arccotan(x)$$ | $$\frac{-1}{x^2+1}$$ |
| $$x.arcsec(x)-ln(x.(1+\sqrt{1-\frac{1}{x^2}}))$$ | $$arcsec(x)$$ | $$\frac{1}{|x|.\sqrt{x^2-1}}$$ |
| $$x.arccosec(x)+ln(x.(1+\sqrt{1-\frac{1}{x^2}}))$$ | $$arccosec(x)$$ | $$\frac{-1}{|x|.\sqrt{x^2-1}}$$ |
| $$cosh(x)$$ | $$sinh(x)$$ | $$cosh(x)$$ |
| $$sinh(x)$$ | $$cosh(x)$$ | $$sinh(x)$$ |
| $$ln(cosh(x))$$ | $$tanh(x)$$ | $$sec^2(x)$$ |
| $$ln(|sinh(x)|)$$ | $$cotanh(x)$$ | $$-cotech^2(x)$$ |
| $$arctan(sinh(x))$$ | $$dry(x)$$ | $$-sec(x).tanh(x)$$ |
| $$ln|tanh(\frac{x}{2}|$$ | $$cotech(x)$$ | $$-cotech(x).cotanh(x)$$ |
| $$x.arg sinh(x)-\sqrt{1+x^2}$$ | $$arg sinh(x)$$ | $$\frac{1}{\sqrt{x^2+1}}$$ |
| $$x.arg cosh(x)-\sqrt{x^2-1}$$ | $$arg cosh(x)$$ | $$\frac{1}{\sqrt{x^2-1}}$$ |
| $$x.arg tanh(x)+\frac{1}{2}.ln(1-x^2)$$ | $$arg tanh(x)$$ | $$\frac{1}{1-x^2}$$ |
| $$x.arg cotanh(x)+\frac{1}{2}.ln(x^2-1)$$ | $$arg cotanh(x)$$ | $$\frac{1}{1-x^2}$$ |
| $$x.arg sech(x)-arctan(\sqrt{\frac{1}{x^2}-1}))$$ | $$arg sec(x)$$ | $$\frac{-1}{x.\sqrt{1-x^2}}$$ |
| $$x.arg cosec(x)+ln(x.(1+\sqrt{1+\frac{1}{x^2}}))$$ | $$arg cosec(x)$$ | $$\frac{-1}{x.\sqrt{1-x^2}}$$ |