To determine

To evaluate:

The given integral.

Answer

12ln⁡x2+4+c

Explanation

1) Concept:

The substitution rule: If u=g(x) is a differentiable function whose range is I,  and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here g(x) is substituted as u and then

g’(x)dx =du

2) Formula:

i.

∫cf(x)dx=c∫f(x)dx

ii.

∫1xdx=ln⁡|x|+c

3) Given:

∫xx2+4 dx

4) Calculation:

Consider the given integral ∫xx2+4 dx

Here using the substitution method

Substitute x2+4=u,

Differentiating with respect to x

2x dx=du

 xdx=du2

Using this in given integral, it becomes

∫xx2+4 dx= ∫1udu2

By taking the constant term outside the integral

=12∫1udu

By using rule of indefinite integral

=12ln⁡|u|+c

Resubstituting u=x2+4 in the above solution

=12ln⁡|x2+4|+c

Since always  x2+4>0  so,

=12ln⁡x2+4 +c

Conclusion:

Therefore, ∫xx2+4 dx= 12ln⁡x2+4+c