Question

integrate 1/1+x^2 dx

Answer 1

arctan(x) + C

Answer 2

that i know.. how u got there is what i want to know

Answer 3

let y = arctan(x)

then x = tan(y)

by implicit differentiation, sec^2(y) * dy/dx = 1

thus dy/dx = cos^2(y)

since x = tan(y), then cos(y) = 1/sqrt(1+x^2)

thus cos^2(y) = 1/(1 + x^2)

thus dy/dx = 1/(1 + x^2)

hence the derivative of arctan(x) is 1/(1 + x^2)

hence the integral of 1/(1 + x^2) is arctan(x)

Answer 4

You could do it with trigonometric substitution. But if you haven't covered that then james^^ has about the best explanation.

Answer 5

The basic trig inverse functions, sin, cos, tan, etc have a certain form. You need to become familiar with these and be able to recognize them.