Solve the Integral of (1+x^2)^(1/2) dx Using Substitution and Integration by Parts

To integrate ( (1+x^2)^(1/2) dx ), we can use the substitution method.

Let's make the substitution: ( x = tan(u) ). Then, ( dx = sec^2(u) du ).

Substituting these values into the integral, we have:

( \int (1+x^2)^(1/2) dx = \int (1+tan^2(u))^(1/2) sec^2(u) du )

Using the trigonometric identity: ( 1 + tan^2(u) = sec^2(u) ), we can simplify the integrand:

( \int (1+tan^2(u))^(1/2) sec^2(u) du = \int (sec^2(u))^(1/2) sec^2(u) du )

Simplifying further:

( = \int sec^3(u) du )

Now, we can use integration by parts to find the integral of ( sec^3(u) ):

Let ( u = sec(u) ) and ( dv = sec^2(u) du ) Then, ( du = sec(u) tan(u) du ) and ( v = tan(u) )

Applying the formula for integration by parts:

( \int sec^3(u) du = sec(u) tan(u) - \int tan^2(u) sec(u) du )

Using the trigonometric identity: ( tan^2(u) = sec^2(u) - 1 ), we can simplify the integrand:

( = sec(u) tan(u) - \int (sec^2(u) - 1) sec(u) du ) ( = sec(u) tan(u) - \int sec^3(u) du + \int sec(u) du )

Now, we have another integral of ( sec(u) ), which can be solved using a similar process:

Let ( u = sec(u) ) and ( dv = du ) Then, ( du = sec(u) tan(u) du ) and ( v = u )

Applying integration by parts again:

( \int sec(u) du = sec(u) u - \int u tan(u) du ) ( = sec(u) u - \int u sec^2(u) du )

We can simplify the integral further:

( = sec(u) u - \int u (1 + tan^2(u)) du ) ( = sec(u) u - \int u (1 + sec^2(u) - 1) du ) ( = sec(u) u - \int u sec^2(u) du + \int u du ) ( = sec(u) u - \int u sec^2(u) du + 0.5u^2 + C )

Now, we can substitute these results back into the original integral:

( \int (1+x^2)^(1/2) dx = sec(u) tan(u) - \int sec^3(u) du + \int sec(u) du ) ( = sec(u) tan(u) - (sec(u) u - \int u sec^2(u) du + 0.5u^2) + 0.5u^2 + C ) ( = sec(u) tan(u) - sec(u) u + \int u sec^2(u) du + 0.5u^2 + C ) ( = sec(u) (tan(u) - u) + \int u sec^2(u) du + 0.5u^2 + C )

Finally, substituting back ( u = tan^(-1)(x) ):

( \int (1+x^2)^(1/2) dx = sec(tan^(-1)(x)) (tan(tan^(-1)(x)) - tan^(-1)(x)) + \int tan^(-1)(x) sec^2(tan^(-1)(x)) dx + 0.5(tan^(-1)(x))^2 + C )

Simplifying the trigonometric functions:

( = (1+x^2)^(1/2) (x - tan^(-1)(x)) + \int tan^(-1)(x) (1 + x^2) dx + 0.5(tan^(-1)(x))^2 + C )

This is the final result of the integral.