Integrals of Trigonometric Functions and Monomials

Indefinite integrals of products of a monomial and a trigonometric function.

\int x cos x d x = cos x + x sin x + C

\int x cos a x d x = \frac{1}{a ^{2}} cos a x + \frac{x}{a} sin a x + C

\int x^{2} cos x d x = 2 x cos x + (x^{2} - 2) sin x + C

\int x^{2} cos a x d x = \frac{2 x cos a x}{a ^{2}} + \frac{a ^{2} x ^{2} - 2}{a ^{3}} sin a x + C

\int x^{n} cos x d x = - \frac{1}{2} (i)^{n + 1} [Γ (n + 1, - i x) + (- 1)^{n} Γ (n + 1, i x)] + C

\int x^{n} cos a x d x = \frac{1}{2} (ia)^{1 - n} [(- 1)^{n} Γ (n + 1, - ia x) - Γ (n + 1, ia x)] + C

\int x sin x d x = - x cos x + sin x + C

\int x sin a x d x = - \frac{x cos a x}{a} + \frac{sin a x}{a ^{2}} + C

\int x^{2} sin x d x = (2 - x^{2}) cos x + 2 x sin x + C

\int x^{2} sin a x d x = \frac{2 - a ^{2} x ^{2}}{a ^{3}} cos a x + \frac{2 x sin a x}{a ^{2}} + C

\int x^{n} sin x d x = - \frac{1}{2} (i)^{n} [Γ (n + 1, - i x) - (- 1)^{n} Γ (n + 1, i x)] + C

\int x cos^{2} x d x = \frac{x ^{2}}{4} + \frac{1}{8} cos 2 x + \frac{1}{4} x sin 2 x + C

\int x sin^{2} x d x = \frac{x ^{2}}{4} - \frac{1}{8} cos 2 x - \frac{1}{4} x sin 2 x + C

\int x tan^{2} x d x = - \frac{x ^{2}}{2} + ln cos x + x tan x + C

\int x sec^{2} x d x = ln cos x + x tan x + C