Question

22. Integrate :
dx
(i)
dx/(1+x^2/4) ​

Answer 1

Answer:

Our goal should be to make this mirror the arctangent integral:

int1/(u^2+1)du=arctan(u)+C

To get the 1 in the denominator, start by factoring:

int1/(x^2+4)dx=int1/(4(x^2/4+1))dx=1/4int1/(x^2/4+1)dx

Note that we want u^2=x^2/4, so we let u=x/2, which implies that du=1/2dx.

1/4int1/(x^2/4+1)dx=1/2int(1/2)/((x/2)^2+1)dx=1/2int1/(u^2+1)du

This is the arctangent integral:

1/2int1/(u^2+1)du=1/2arctan(u)+C=1/2arctan(x/2)+C

Step-by-step explanation: