#arctan

What’s Your Angle?

A mathcomic for Sophia Woods’ #mathober theme “angle.”  I always wished we learned more trig values than the standard table/unit circle.

28. Integration of 1 / (x^5 + 1)

Question. (a) By using partial fractions, show that where (b) Hence, or otherwise, show that where (c) Hence, or otherwise, show that Solution. (a) (1) Factorisation of the denominator For the second factor, we seem to have two options: The first line gives while the second line yields Since the second line gives two complex conjugate roots, we select the first one and…

27. Integration of 1 / (x^4 + 1)

Question. (a) By using partial fractions, show that (b) Hence, or otherwise, show that (c) Hence, or otherwise, show that Solution. (a) (1) Factorisation of the denominator: Can think of two possibilities: For real values of , we have to choose the first line with (the second line gives complex values for ): i.e. (2) Partial fractions Solving four simultaneous equations gives Finally, (b)…

35. Integration of 1/(x^3+1)

Question. (a) By using partial fractions, show that (b) Hence, or otherwise, show that (c) Hence, or otherwise, show that (a) (1) Factorisation of the denominator (2) By partial fractions, By the method of substitution (can equally use the method of equating the coefficients) which gives Hence, (b) Integration For the last part, we use the following substitution: which gives Finally, we…

5. Differentiation – Inverse trigonometric functions

Question. Find for:(a) (b) Solution. (a) For , we have We re-express the equation By implicit differentiation, (b) For , we have We re-express the equation By chain rule,

The Lingual Anti-Derivative :P #craton