Calculus Formula Flashcards

Flashcards covering fundamental calculus formulas for differentiation and integration based on the transcript.

Derivative of a Constant

$$\frac{d}{dx}(k) = 0$$

Derivative of $e^x$

$$\frac{d}{dx}(e^x) = e^x$$

Derivative of $e^x$

$$\frac{d}{dx}(e^x) = e^x$$

Derivative of $a^x$

$$\frac{d}{dx}(a^x) = a^x \ln(a)$$

Derivative of $\ln(x)$

$$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$

Derivative of $\log_a(x)$

$$\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$$

Derivative of $\sin(x)$

$$\frac{d}{dx}(\sin(x)) = \cos(x)$$

Derivative of $\cos(x)$

$$\frac{d}{dx}(\cos(x))=-\sin(x)$$

Derivative of $\tan(x)$

$$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

Derivative of $\csc(x)$

$$\frac{d}{dx}(\csc(x))=-\csc(x)\cot(x)$$

Derivative of $\sec(x)$

$$\frac{d}{dx}(\sec(x)) = \sec(x) \tan(x)$$

Derivative of $\cot(x)$

$$\frac{d}{dx}(\cot(x)) = -\csc^2(x)$$

Derivative of $\sin^{-1}(x)$

$$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$

Derivative of $\cos^{-1}(x)$

$$\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^2}}$$

Derivative of $\tan^{-1}(x)$

$$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$$

Integral of a Constant

$$\int k\,dx = kx + C$$

Power Rule for Integration

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$

Integral of $\frac{1}{x}$

$$\int \frac{1}{x}\,dx = \ln|x| + C$$

Integral of $e^x$

$$\int e^x\,dx = e^x + C$$

Integral of $a^x$

$$\int a^x\,dx = \frac{a^x}{\ln(a)} + C$$

Integral of $\sin(x)$

$$\int\sin(x)\,dx=-\cos(x)+C$$

Integral of $\cos(x)$

$$\int \cos(x)\,dx = \sin(x) + C$$

Integral of $\sec^2(x)$

$$\int \sec^2(x)\,dx = \tan(x) + C$$

Integral of $\csc^2(x)$

$$\int \csc^2(x)\,dx = -\cot(x) + C$$

Integral of $\sec(x)\tan(x)$

$$\int \sec(x)\tan(x)\,dx = \sec(x) + C$$

Integral of $\csc(x)\cot(x)$

$$\int \csc(x)\cot(x)\,dx = -\csc(x) + C$$

Integral of $\tan(x)$

$$\int \tan(x)\,dx = \ln|\sec(x)| + C$$

Integral of $\cot(x)$

$$\int \cot(x)\,dx = \ln|\sin(x)| + C$$

Integral of $\sec(x)$

$$\int \sec(x)\,dx = \ln|\sec(x) + \tan(x)| + C$$

Integral of $\csc(x)$

$$\frac{d}{dx}\int \csc(x)\,dx = -\ln|\csc(x) + \cot(x)| + C$$

Integral of $\frac{1}{\sqrt{a^2-x^2}}$

$$\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C$$

Integral of $\frac{1}{\sqrt{1-x^2}}$

$$\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin(x) + C$$

Integral of $\frac{1}{a^2+x^2}$

$$\int \frac{1}{a^2+x^2}\,dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$

Integral of $\frac{1}{1+x^2}$

$$\int \frac{1}{1+x^2}\,dx = \arctan(x) + C$$