Integral Calculus formulas

∫ c dx = cx + C

The integral of a constant 'c' with respect to 'x' is equal to 'cx' plus a constant 'C'.

∫ x^n dx = (x⁽ⁿ⁺¹⁾)/(n+1) + C, (n ≠ -1)

The integral of 'x' raised to the power of 'n' with respect to 'x' is equal to '(x^(n+1))/(n+1)' plus a constant 'C' (where 'n' is not equal to -1).

∫ e^x dx = e^x + C

The integral of 'e' raised to the power of 'x' with respect to 'x' is equal to 'e^x' plus a constant 'C'.

∫ e^(kx) dx = (e^(kx))/k + C

The integral of the exponential function 'e' raised to the power of 'kx' with respect to 'x' is equal to '(e^(kx))/k' plus a constant 'C'.

∫ (1/x) dx = ln|x| + C

The integral of '1' divided by 'x' with respect to 'x' is equal to the natural logarithm of the absolute value of 'x' plus a constant 'C'.

∫ sin(x) dx = -cos(x) + C

The integral of the sine function with respect to 'x' is equal to the negative cosine of 'x' plus a constant 'C'.

∫ cos(x) dx = sin(x) + C

The integral of the cosine function with respect to 'x' is equal to the sine of 'x' plus a constant 'C'.

∫ sec^2(x) dx = tan(x) + C

The integral of the secant squared function with respect to 'x' is equal to the tangent of 'x' plus a constant 'C'.

∫ csc^2(x) dx = -cot(x) + C

The integral of the cosecant squared function with respect to 'x' is equal to the negative cotangent of 'x' plus a constant 'C'.

∫ sec(x)tan(x) dx = sec(x) + C

The integral of the secant of 'x' multiplied by the tangent of 'x' with respect to 'x' is equal to the secant of 'x' plus a constant 'C'.

∫ csc(x)cot(x) dx = -csc(x) + C

The integral of the cosecant of 'x' multiplied by the cotangent of 'x' with respect to 'x' is equal to the negative cosecant of 'x' plus a constant 'C'.

∫ 1/(√(1-x^2)) dx = arcsin(x) + C

∫ 1/(1+x^2) dx = arctan(x) + C

∫ 1/|x|(√(x^2-1)) dx = arcsec|x| + C

∫ u dv = uv - ∫ v du

∫ √(a^2 - x^2) dx = (1/2) [x√(a^2 - x^2) + a^2 arcsin(x/a)] + C

The integral of the square root of 'a^2 - x^2' with respect to 'x' can be evaluated using a trigonometric substitution.

∫ √(x^2 + a^2) dx = (1/2) [x√(x^2 + a^2) + a^2 ln(x + √(x^2 + a^2))] + C

The integral of the square root of 'x^2 + a^2' with respect to 'x' can be evaluated using a trigonometric substitution.

∫ √(x^2 - a^2) dx = (1/2) [x√(x^2 - a^2) - a^2 ln(x + √(x^2 - a^2))] + C

The integral of the square root of 'x^2 - a^2' with respect to 'x' can be evaluated using a trigonometric substitution.

∫ (Ax + B)/(x^2 + px + q) dx = (A/2) ln|x^2 + px + q| + (2B-Ap)/√(4p-p^2) arctan((2x+p)/√(4p-p^2)) + C

The integral of a rational function with respect to 'x' can be evaluated using partial fraction decomposition.

∫ f(g(x)) * g'(x) dx = ∫ f(u) du

∫_a^b f(x) dx = F(b) - F(a)

∫ sin(x)cos(x) dx = (1/2)sin^2(x) + C

∫ cos^2(x) dx = (1/2)(x + sin(x)cos(x)) + C

∫ sec(x)tan(x) dx = sec(x) + C