Table 8.1 Basic Integration Formulas

Basic Integration Formulas

∫k dx

Integral of k dx equals kx + C, where k is a real number.

∫x^p dx

Integral of x^p dx equals (x^(p+1))/(p+1) + C, where p ≠ -1 is a real number.

∫cos(ax) dx

Integral of cos(ax) dx equals (1/a)sin(ax) + C.

∫sin(ax) dx

Integral of sin(ax) dx equals (-1/a)cos(ax) + C.

∫sec²(ax) dx

Integral of sec²(ax) dx equals (1/a)tan(ax) + C.

∫tan(ax) dx

Integral of tan(ax) dx equals (-1/a)ln|cos(ax)| + C or (1/a)ln|sec(ax)| + C.

∫sec(ax)tan(ax) dx

Integral of sec(ax)tan(ax) dx equals (1/a)sec(ax) + C.

∫csc(ax)cot(ax) dx

Integral of csc(ax)cot(ax) dx equals (-1/a)csc(ax) + C.

∫e^(ax) dx

Integral of e^(ax) dx equals (1/a)e^(ax) + C.

∫(1/x) dx

Integral of 1/x dx equals ln|x| + C.

∫1/(a² + x²) dx

Integral of 1/(a² + x²) dx equals (1/a)tan⁻¹(x/a) + C, where a > 0.

∫1/(x√(x² - a²)) dx

Integral of 1/(x√(x² - a²)) dx equals (1/a)sec⁻¹(x/a) + C, where a > 0.

∫1/√(a² - x²) dx

Integral of 1/√(a² - x²) dx equals sin⁻¹(x/a) + C, where a > 0.

∫cot(ax) dx

Integral of cot(ax) dx equals (1/a)ln|sin(ax)| + C.

∫sec(ax) dx

Integral of sec(ax) dx equals (1/a)ln|sec(ax) + tan(ax)| + C.

∫csc(ax) dx

Integral of csc(ax) dx equals (-1/a)ln|csc(ax) + cot(ax)| + C.