#### 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-1x+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-1u+c

Resubstituting u=x2 intheabove solution

=12tan-1(x2) +c

**Conclusion:**

Therefore, ∫x1+x4dx=12tan-1x2+c