Table of Integrals

Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. Here is the beginning of such a table.

We use c_a for an arbitrary constant that can only be determined if something about the value of the integral at some point is known.

∫xⁿ dx = xⁿ⁺¹/(n+1) + c_a    (n ≠ -1)

∫x^-1 dx = ln(|x|) + c_a

---

∫ln(x) dx = x ln(x) - x + c_a

---

∫e^x dx = e^x + c_a

∫a^x dx = a^x/ln(a) + c_a

---

∫(1+x²)^-1 dx = arctan(x) + c_a

-∫(1+x²)^-1 dx = arccot(x) + c_a

∫(1-x²)^-1/2 dx = arcsin(x) + c_a

-∫(1-x²)^-1/2 dx = arccos(x) + c_a

∫x(x²-1)^-1/2 dx = arcsec(x) + c_a

-∫x(x²-1)^-1/2 dx = arccsc(x) + c_a

---

∫cos(x) dx = sin(x) + c_a

∫sin(x) dx = -cos(x) + c_a

∫tan(x) dx = -ln|cos(x)| + c_a

∫csc(x) dx = -ln|csc(x)+cot(x)| + c_a

∫sec(x) dx = ln|sec(x)+tan(x)| + c_a

∫cot(x) dx = ln|sin(x)| + c_a

---

∫sec²(x) dx = tan(x) + c_a

∫csc²(x) dx = -cot(x) + c_a

∫sin²(x) dx = x/2-(sin(2x))/4 + c_a

∫cos²(x) dx = x/2+(sin(2x))/4 + c_a

---

∫sinh(x) dx = cosh(x) + c_a

∫cosh(x) dx = sinh(x) + c_a

∫tanh(x) dx = ln(cosh(x)) + c_a

∫csch(x) dx = ln|tanh(x/2)| + c_a

∫sech(x) dx = arctan(sinh(x)) + c_a

∫coth(x) dx = ln|sinh(x)| + c_a