Questions: Find the antiderivative for each function when C equals 0. Do as many as you can mentally. Check your answers by differentiation. a. 5x^4 b. x^9 c. x^2-9x+18 a. The antiderivative of 5x^4 is

Solution Steps

To find the antiderivative of a polynomial function, we use the power rule for integration. This involves increasing the exponent by one and dividing by the new exponent. Since the constant of integration \( C \) is given as zero, we omit adding any constant term.

The antiderivative of \( 5x^4 \) is calculated as follows:

\[ \int 5x^4 \, dx = \frac{5}{5}x^{4+1} = x^5 \]

The antiderivative of \( x^9 \) is:

\[ \int x^9 \, dx = \frac{1}{10}x^{9+1} = \frac{1}{10}x^{10} \]

The antiderivative of \( x^2 - 9x + 18 \) is:

\[ \int (x^2 - 9x + 18) \, dx = \frac{1}{3}x^{3} - \frac{9}{2}x^{2} + 18x \]

To confirm the antiderivatives, we differentiate each result and check if we obtain the original function.

\[ \frac{d}{dx}(x^5) = 5x^4 \]

This matches the original function \( 5x^4 \).

\[ \frac{d}{dx}\left(\frac{1}{10}x^{10}\right) = x^9 \]

This matches the original function \( x^9 \).

\[ \frac{d}{dx}\left(\frac{1}{3}x^{3} - \frac{9}{2}x^{2} + 18x\right) = x^2 - 9x + 18 \]

This matches the original function \( x^2 - 9x + 18 \).

Final Answer