Evaluate Integrals Involving Quadratics Using Completing Square

Tutorials with examples and detailed solutions and exercises with answers on how to use the techniques of completing square and substitution to evaluate integrals.

Review:

We first review some of the derivatives formulas for known inverse functions.

a) d(arcsin(x)) / dx = 1 / sqrt(1 - x^{2})

b) d(arctan(x)) / dx = 1 / (1 + x^{2})

c) d(arcsinh(x)) / dx = 1 / sqrt(1 + x^{2})

d) d(arccosh(x)) / dx = 1 / sqrt(x^{2} - 1)

We now use the above differentiation formulas to write integrals as follows.

NOTE: in what follows, K is the constant of integration.

1) [ 1 / sqrt(1 - x^{2}) ] dx = arcsin(x) + K

2) [ 1 / (1 + x^{2}) ] dx = arctan(x) + K

3) [ 1 / sqrt(1 + x^{2}) ] dx = arcsinh(x) + K

4) [ 1 / sqrt(x^{2} - 1) ] dx = arccosh(x) + K

Finally one more integral is computed using partial fractions decomposition as follows