Anti-differentiation Formulas

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General Formulas

Constant Multiple Rule: For any constant c and any quantity u, d(cu)=cdu.
The anti-derivative of a constant times a quantity is the constant times the anti-derivative of the quantity.

Sum Rule: d(u + v) = du + dv.
The anti-derivative of the sum of two quantities is the sum of their anti-derivatives.

Integration by Parts: udv = d(uv) - vdu.
The anti-derivative of the product of a first quantity and the differential of a second quantity is the product of the two quantities minus the anti-derivative of the product of the second quantity and the differential of the first quantity.

Power Rule: For any (rational) constant n not equal to -1 and any quantity u, d(u^(n + 1)/(n + 1)) = (u^n)du.
The anti-derivative of a quantity raised to a power (other than -1.) times the differential of that quantity is the quantity raised to the original power plus1, divided by that new power.

Trigonometric Formulas

d(sin(u)) = cos(u)du
The anti-derivative of the cosine of a quantity times the differential of the quantity is the sine of the quantity.

d(-cos(u)) = sin(u)du
The anti-derivative of the sine of a quantity times the differential of the quantity is the negative of the cosine of the quantity.

d(tan(u)) = (sec(u))^2du
The anti-derivative of the square of the secant of a quantity times the differential of the quantity is the tangent of the quantity.

d(sec(u)) = sec(u)tan(u)du
The anti-derivative of the product of the secant of a quantity, the tangent of the quantity, and the differential of the quantity is the secant of the quantity.

d((2u - sin(2u))/4) = (sin(u))^2
The anti-derivative of the square of the sine of a quantity is twice the quantity minus the sine of twice the quantity, all divided by 4.

d((2u + sin(2u))/4) = (cos(u))^2
The anti-derivative of the square of the cosine of a quantity is twice the quantity plus the sine of twice the quantity, all divided by 4.

Inverse Trigonometric Formulas

d(arctan(u)) = du/(1 + u^2)
The anti-derivative of the reciprocal of 1 plus the square of a quantity, times the differential of the quantity, is the inverse tangent of the quantity.

d(arcsin(u)) = du/sqrt(1 - u^2)
The anti-derivative of the reciprocal of the square root of 1 minus the square of a quantity, times the differential of the quantity, is the inverse sine of the quantity.

d(arcsec(u)) = du/(u*sqrt(u^2 - 1))
The anti-derivative of the reciprocal of the product of a quantity and the square root of the square of the quantity minus 1, times the differential of the quantity, is the inverse secant of the quantity.

Exponential Function Formulas

d(e^u) = (e^u)du
The anti-derivative of e raised to a quantity times the differential of the quantity is e raised to that quantity.

d(e^(-u)) = (-e^(-u))du
The anti-derivative of e raised to the negation of a quantity times the differential of the quantity is the negation of e raised to that negation of the quantity.

For any positive base b, (b not equal to 1), d(b^u/ln(b)) = (b^u)du
The anti-derivative of b raised to a quantity divided by the natural logarithm of b is b raised to that quantity times the differential of the quantity.

Logarithm Function Formula

d(ln(u)) = du/u
The anti-derivative of the reciprocal of a quantity times the differential of the quantity is the natural logarithm of the quantity.

Hyperbolic Function Formulas

d(cosh(u)) =sinh(u)du
The anti-derivative of the hyperbolic sine of a quantity times the differential of the quantity is the hyperbolic cosine of the quantity.

d(sinh(u)) =cosh(u)du
The anti-derivative of the hyperbolic cosine of a quantity times the differential of the quantity is the hyperbolic sine of the quantity.

Inverse Hyperbolic Function Formulas

d(arcsinh(u)) = du/sqrt(1 + u^2)
The anti-derivative of the reciprocal of the square root of 1 plus the square of a quantity, times the differential of the quantity, is the inverse hyperbolic sine of the quantity.

d(arccosh(u)) = du/sqrt( u^2 - 1)
The anti-derivative of the reciprocal of the square root of the square of a quantity minus 1, times the differential of the quantity, is the inverse hyperbolic cosine of the quantity.

d(-arcsech(u)) = du/(u*sqrt(1 - u^2))
The anti-derivative of the reciprocal of the product of a quantity and the square root of 1 minus the square of the quantity, times the differential of the quantity, is the inverse hyperbolic secant of the quantity.

d(arctanh(u)) = du/(1 - u^2)
The anti-derivative of the reciprocal of 1 minus the square of a quantity, times the differential of the quantity, is the inverse hyperbolic tangent of the quantity.