You are here:

Master Calculus! Get instant help on “I Aced Calculus AP” App. Hundreds of flashcards and practice questions at your fingertips. Download now on the App Store and Google Play.

Integration Formula Sheet

Fundamental Theorem of Calculus:

Part one:
$$\int_a^bf(x)\:dx=F(b)-F(a)$$

Where $F(x)$ is an antiderivative of $f(x)$

Part two:
$$\dfrac{d}{dx}\int^x_af(t)\:dt=f(x)$$

Basic Integration Rules:

Constant Rule: $\displaystyle{\int} 0 \:dx =C$

Constant Multiple Rule: $\displaystyle{\int} Cf(x) \:dx = C\displaystyle{\int}f(x) \:dx$

Sum and Difference Rule: $\displaystyle{\int}\left[ f(x)\pm g(x) \right]\:dx = \displaystyle{\int}f(x) \:dx\pm \displaystyle{\int}g(x)\:dx$
Power Rule: $\displaystyle{\int}x^n \:dx= \dfrac{x^{n+1}}{n+1}+C$

Inverse Bounds Rule: $\displaystyle{\int^b_a}f(x)\:dx=-\displaystyle{\int^a_b}f(x)\:dx$
Interval Addition Rule: $\displaystyle{\int^b_a}f(x)\:dx+\displaystyle{\int_b^c}f(x)\:dx=\displaystyle{\int^c_a}f(x)\:dx$

Equal Limits Rule: $\displaystyle{\int^a_a}f(x)\:dx=0$

Basic Integration Techniques:

Integration by Parts:

$$\int u(x)v'(x)\:dx=u(x)v(x)-\int v(x)v'(x)\:dx$$

Integration by Substitution:

$$\int f(g(x))g'(x)\:dx= \int f(u)\:du$$

Where $u=g(x)$ and $du = g'(x)\:dx$

Exponential and Logarithmic Integrals

$$\begin{array}{lll} \displaystyle{\int} e^x \: dx = e^x + C && \displaystyle{\int} a^x \: dx = \dfrac{a^x}{\ln{a}} + C \\[10pt] \displaystyle{\int} \dfrac{1}{x} \: dx = \ln{|x|} + C && \displaystyle{\int} \dfrac{1}{x \ln{a}} \: dx = \log_a |x| + C \end{array}$$

Trigonometric Integrals

\[ \begin{align*} &\hspace{-2cm}\int \sin x \: dx = -\cos x + C &\int \sec x \tan x \: dx = \sec x + C \\ &\hspace{-2cm}\int \cos x \: dx = \sin x + C &\int \csc x \cot x \: dx = -\csc x + C \\ &\hspace{-2cm}\int \tan x \: dx = -\ln |\cos x| + C & \int \cot x \: dx = \ln |\sin x| + C\\ &\hspace{-2cm}\int \sec^2 x \: dx = \tan x + C & \int \csc^2 x \: dx = -\cot x + C\\ \end{align*} \]

Inverse Trigonometric Integrals

\[ \begin{align*} &\hspace{-0.5cm}\int \sin^{-1} x \: dx = x \sin^{-1} x + \sqrt{1-x^2} + C &\int \sec^{-1} x \: dx = x \sec^{-1} x – \ln |x + \sqrt{x^2 – 1}| + C \\ &\hspace{-.5cm}\int \cos^{-1} x \: dx = x \cos^{-1} x + \sqrt{1-x^2} + C &\int \csc^{-1} x \: dx = x \csc^{-1} x – \ln |x + \sqrt{x^2 – 1}| + C \\ &\hspace{-.5cm}\int \tan^{-1} x \: dx = x \tan^{-1} x – \dfrac{1}{2} \ln(x^2 + 1) + C & \int \cot^{-1} x \: dx = x \cot^{-1} x + \dfrac{1}{2} \ln(x^2 + 1) + C \end{align*} \]

NEED QUICK
CALC HELP?
Download the I Aced Calculus App today!
ALL Calc Topics, 1000+ of PRACTICE questions