purple sheet

Antiderivative formulas to memorize

xⁿ

xⁿ⁺¹/n+1 +C , n cannot = 1

e^kx

1/k (e^kx )+C

n^kx

1/k ln n (n^kx)+C

1/x

ln |x| +C , x cannot be = 0

sin kx

-1/k (cos kx) +C

cos kx

1/k (sin kx) +C

tan kx

1/k (ln |sec kx|) +C

cot kx

1/k (ln |sin kx|) +C

sec kx

1/k (ln |sec kx + tan kx|) +C

csc kx

-1/k (ln |csc kx + cot kx|) +C

sec² kx

1/k (tan kx) +C

csc² kx

-1/k (cot kx) +C

sec kx tan kx

1/k (sec kx) +C

csc kx cot kx

-1/k (csc kx) +C

1/ √n²-x²

sin^-1(x/n) +C

1/n²+x²

1/n (tan^-1(x/n) +C

sec³(x) dx

½ (secx tanx) + ½ ln |secx +tanx| +C

1/ax+b dx

1/a ln |ax+b| +C ,a cannot = 0

The Fundamental Theorem of Calculus

|_a^b f’(x) dx = f(x) |_a^b = f(b) - f(a)

U Substitution

|_a^b f(g(x)) g’(x) dx |_g(a)^g(b) f(u) du , where u= g(x), du= g’(x) dx

Integration by Parts

|_a^b u dv= uv -|_a^b v du