Understand the idea Antidifferentiation reverses derivative rules and includes an arbitrary constant. The power rule for integration excludes exponent − 1 \displaystyle -1 − 1 ∫ csc 2 x d x = − cot x + C \displaystyle \int\csc^2x\,dx=-\cot x+C ∫ csc 2 x d x = − cot x + C ∫ csc x cot x d x = − csc x + C \displaystyle \int\csc x\cot x\,dx=-\csc x+C ∫ csc x cot x d x = − csc x + C ∫ d x / a 2 − x 2 = arcsin ( x / a ) + C \displaystyle \int dx/\sqrt{a^2-x^2}=\arcsin(x/a)+C ∫ d x / a 2 − x 2 = arcsin ( x / a ) + C a > 0 \displaystyle a>0 a > 0 ∣ x ∣ < a \displaystyle |x|<a ∣ x ∣ < a
∫ x n d x = x n + 1 n + 1 + C ( n ≠ − 1 ) ∫ u ′ u d x = ln ∣ u ∣ + C ∫ e x d x = e x + C ∫ sin x d x = − cos x + C ∫ cos x d x = sin x + C ∫ sec 2 x d x = tan x + C ∫ sec x tan x d x = sec x + C \begin{aligned}\int x^n\,dx&=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)\\\int\frac{u'}{u}\,dx&=\ln|u|+C\\\int e^x\,dx&=e^x+C\\\int\sin x\,dx&=-\cos x+C\\\int\cos x\,dx&=\sin x+C\\\int\sec^2x\,dx&=\tan x+C\\\int\sec x\tan x\,dx&=\sec x+C\end{aligned} ∫ x n d x ∫ u u ′ d x ∫ e x d x ∫ sin x d x ∫ cos x d x ∫ sec 2 x d x ∫ sec x tan x d x = n + 1 x n + 1 + C ( n = − 1 ) = ln ∣ u ∣ + C = e x + C = − cos x + C = sin x + C = tan x + C = sec x + C Worked example Evaluate ∫ 0 1 2 x e x 2 d x \displaystyle \int_0^1 2x e^{x^2}\,dx ∫ 0 1 2 x e x 2 d x
Let
u = x 2 \displaystyle u=x^2 u = x 2 , so
d u = 2 x d x \displaystyle du=2x\,dx d u = 2 x d x . The bounds become
0 \displaystyle 0 0 and
1 \displaystyle 1 1 . The integral is
∫ 0 1 e u d u = e − 1 \displaystyle \int_0^1e^u\,du=e-1 ∫ 0 1 e u d u = e − 1 . Differentiation of
e x 2 \displaystyle e^{x^2} e x 2 checks the antiderivative.
Check this before moving on Forgetting + C \displaystyle +C + C − 1 \displaystyle -1 − 1
Try it yourself Find ∫ 3 x 2 / ( x 3 + 5 ) d x \displaystyle \int 3x^2/(x^3+5)\,dx ∫ 3 x 2 / ( x 3 + 5 ) d x
Show solution and reasoning With u = x 3 + 5 \displaystyle u=x^3+5 u = x 3 + 5 ln ∣ x 3 + 5 ∣ + C \displaystyle \ln|x^3+5|+C ln ∣ x 3 + 5∣ + C
Work through AP Calculus AB with a tutor Detailed notes are discussed during the course with worked examples, guided practice, and feedback on mathematical reasoning.
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