Evaluate the following integrals:
The given equation can be written as
⇒
First integration be I1 and second be I2.
⇒ For I1
Add and subtract 2 from the numerator
⇒
⇒
⇒
⇒ x - 2ln|x + 2| + c1
∴ I1 = x - 2ln|x + 2| + c1
For I2
⇒
Assume x + 1 = t
dt = dx
⇒
Substitute u = √t
dt = 2√t.du
t = u2
⇒
Add and subtract 1 in the above equation:
⇒
⇒
⇒
⇒ 2u - tan - 1(u) + c2
But u = √t
∴ 2√t - tan - 1(√t) + c2
Also t = x + 1
∴ 2√(x + 1) - tan - 1(x + 1) + c2
I = I1 + I2
∴ I = x - 2ln|x + 2| + c1 + 2√(x + 1) - tan - 1(x + 1) + c2
I = x - 2ln|x + 2| + 2√(x + 1) - tan - 1(x + 1) + c.