# Further Maths - Integrating and Differentiating Inverse Trig Functions

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/integrating-and-differentiating-inverse-trig-functions/ · Updated: 2021-04-29 · Tags: further-maths, school, methods-in-calculus

## See Also
- [Further Maths - Mean Value of a Function](https://ollybritton.com/notes/a-level/further-maths/topics/mean-value-of-a-function/)
- [Further Maths - Improper Integrals](https://ollybritton.com/notes/a-level/further-maths/topics/improper-integrals/)
- [Further Maths - Hyperbolic Functions](https://ollybritton.com/notes/a-level/further-maths/topics/hyperbolic-functions/)

## Flashcards
##### $$\frac{d}{dx}(\sin^{-1}(x))$$ What is this equal to??
$$
\frac{1}{\sqrt{1 - x^2}}
$$

##### $$\frac{d}{dx}(\cos^{-1}(x))$$ What is this equal to??
$$
-\frac{1}{\sqrt{1 - x^2}}
$$

##### $$\frac{d}{dx}(\tan^{-1}(x))$$ What is this equal to??
$$
\frac{1}{1 + x^2}
$$

##### $$\int \frac{1}{\sqrt{1-x^2}} dx$$ What is this equal to??
$$
\sin^{-1}(x) + c
$$

##### $$\int -\frac{1}{\sqrt{1-x^2}} dx$$ What is this equal to??
$$
\cos^{-1}(x) + c
$$

##### $$\int \frac{1}{1 + x^2}dx$$ What is this equal to??
$$
\tan^{-1}(x) + c
$$

##### $$\int \frac{1}{\sqrt{a^2-x^2}} dx$$ What is this equal to??
$$
\sin^{-1}\left(\frac{x}{a}\right) + c
$$

##### $$\int \frac{1}{a^2 + x^2}dx$$ What is this equal to??
$$
\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + c
$$

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