To determine

To evaluate:

The given integral.

Answer

12tan-1⁡(x2)+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:

∫11+x2dx=tan-1⁡x+c

3) Given:

∫x1+x4dx

4) Calculation:

Consider the given integral ∫x1+x4dx

Here using the substitution method

Substitute x2=u,

Differentiating with respect to x

2x dx=du

xdx=12du,

Using this in the given integral, it becomes

∫x1+x4dx=∫11+u2du2

By taking the constant term outside the integral

=12∫11+u2 du

By using the rule of indefinite integral

=12tan-1⁡u+c

Resubstituting u=x2 intheabove solution

=12tan-1⁡(x2) +c

Conclusion:

Therefore, ∫x1+x4dx=12tan-1⁡x2+c