#### To determine

**To evaluate:**

The integral ∫x3x4-52 dx by making the given substitution u=x4-5.

#### Answer

-14(x4-5)+C

#### Explanation

**1) Concept:**

i) The substitution rule is that

If u=g(x) is a differentiable function whose range is an interval I and f is continuous oninterval I, then ∫f(gx)g'xdx=∫f(u)du.

ii) Indefinite integral

∫xn dx=xn+1n+1+C (n≠-1)

**2) Given:**

∫x3x4-52 dx u=x4-5

**3) Calculation:**

Use the substitution u=x4-5

Differentiate u=(x4-5) with respect to x

du=4x3dx

As x3 dx is a part of the integration, solving for x3 dx by dividing both side by 4.

x3dx=du4

By using concept i)

Substitute u=x4-5,x3dx=du4 to get ∫x3x4-52 dx=∫1u2du4

By using concept ii)

=14u-2+1-2+1+C

By addition

=14u-1-1+C

=-14u+C

Resubstitute u=x4-5.

=-14(x4-5)+C

**Conclusion:**

Therefore,

∫x3x4-52 dx=-14(x4-5)+C