Let's find the antiderivatives for each function given, assuming $C = 0$. This means we’ll ignore the constant of integration.

The power rule for antiderivatives states that if $f(x) = x^n$, then: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ when $n \neq -1$.

Solutions:

a. $f(x) = 5x^4$

For $5x^4$, increase the exponent by 1 and divide by the new exponent: $\int 5x^4 \, dx = 5 \cdot \frac{x^{4+1}}{4+1} = 5 \cdot \frac{x^5}{5} = x^5$

b. $f(x) = x^{11}$

For $x^{11}$, apply the power rule: $\int x^{11} \, dx = \frac{x^{11+1}}{11+1} = \frac{x^{12}}{12}$

c. $f(x) = x^2 - 8x + 15$

We can take the antiderivative term by term:

1. For $x^2$: $\int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}$
2. For $-8x$: $\int -8x \, dx = -8 \cdot \frac{x^{1+1}}{1+1} = -8 \cdot \frac{x^2}{2} = -4x^2$
3. For $15$: $\int 15 \, dx = 15x$

So, combining these: $\int \left( x^2 - 8x + 15 \right) dx = \frac{x^3}{3} - 4x^2 + 15x$

Summary of Antiderivatives:

• a. $x^5$
• b. $\frac{x^{12}}{12}$
• c. $\frac{x^3}{3} - 4x^2 + 15x$

Would you like me to verify these results by differentiation or provide further details?

Related Questions:

1. What is the antiderivative of $x^{n}$ in general?
2. How would the answer change if $C \neq 0$?
3. What is the role of constants in antiderivatives?
4. How can you verify an antiderivative through differentiation?
5. How does the power rule change when $n = -1$?

Tip:

When finding antiderivatives, always add $C$ unless specifically told not to, as it represents the family of all antiderivatives.