To determine
To evaluate:
The given integral.
Answer
12lnx2+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,
=12lnx2+4 +c
Conclusion:
Therefore, ∫xx2+4 dx= 12lnx2+4+c