Calculus: Derivatives and Integration Rules

Vocabulary and formula-based flashcards covering integration rules, derivatives of trigonometric and hyperbolic functions, and techniques like u-substitution and conjugates.

$$\frac{d}{dx} b^x$$

$$b^{x}\text{ln}(b)$$

$$\int b^x dx$$

$$\frac{b^x}{\text{ln}(b)}+C$$

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

$$-\sin(x)$$

$$\int \sin(x) dx$$

$$-\cos(x) + C$$

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

$$\cos(x)$$

$$\int \cos(x) dx$$

$$\sin(x) + C$$

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

$$\sec^2(x)$$

$$\int \sec^2(x) dx$$

$$\tan(x) + C$$

$$\frac{d}{dx} \sinh(x)$$

$$\cosh(x)$$

$$\int \cosh(x) dx$$

$$\sinh(x) + C$$

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

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

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

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

$$\int \tan(x) dx$$

$$\ln\left|\sec(x)\left|+C\right.\right.$$

$$\int \sec(x) dx$$

$$\ln\left|\sec(x)+\tan(x)\right|+C$$

$$\int \csc(x) dx$$

$$-\ln\left|\csc(x)+\cot(x)\right|+C$$

$$\int e^{ax} dx$$

$$\frac{1}{a} e^{ax} + C$$

$$\int \tanh(x) dx$$

$$\ln(\cosh(x)) + C$$

$$\int \sin^{-1}(x) \, dx$$

$$x \sin^{-1}(x) + \sqrt{1-x^2} + C$$

$$\int \frac{1}{a^2+x^2} dx$$

$$\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$$