#### To determine

**To evaluate:**

∫011-x9dx if it exists

#### Answer

110

#### Explanation

**1) Concept:**

By using the substitution rule for definite integrals and fundamental rule of calculus

**2) Theorem and Rule:**

Fundamental theorem of calculus:

If f is continuous on [a, b], then ∫abfxdx=Fb-F(a).

Substitution rule for definite integrals:

If g' is continuous on [a, b] and f is continuous on the range of u=g(x) then

∫abfgxg'xdx=∫g(a)g(b)f(u)du

**3) Formula:**

∫xndx=xn+1n+1+C

**3) Given:**

∫011-x9dx

**4) Calculation:**

Consider, ∫011-x9dx

Since the function is continuous in the given interval the integral exists.

By applying the substitution rule for definite integrals

Let u=1-x. Then du=-dx, so dx=-du

When x=0, u=1 and x=1, u=0

Thus,

∫011-x9dx=-∫10u9du

By using the fundamental theorem of calculus and power rule,

=-u101010

Simplifying,

=-[01010-11010]

=-(-110)

=110

**Conclusion:**

Therefore, ∫011-x9dx=110