find the antiderivative for each function when c equals 0. do as many as you can mentally. check your…

Explanation:

Step1: Recall power - rule for antiderivatives

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$).

Step2: Find antiderivative of $a.6x^{5}$

Using the constant - multiple rule and power - rule, if $y = 6x^{5}$, then $\int6x^{5}dx=6\times\frac{x^{5 + 1}}{5+1}=x^{6}$.

Step3: Find antiderivative of $b.x^{4}$

Applying the power - rule, if $y=x^{4}$, then $\int x^{4}dx=\frac{x^{4 + 1}}{4+1}=\frac{1}{5}x^{5}$.

Step4: Find antiderivative of $c.x^{2}-10x + 24$

Using the sum - difference rule and power - rule: $\int(x^{2}-10x + 24)dx=\int x^{2}dx-10\int xdx+24\int dx$. $\int x^{2}dx=\frac{x^{3}}{3}$, $\int xdx=\frac{x^{2}}{2}$, $\int dx=x$. So $\int(x^{2}-10x + 24)dx=\frac{1}{3}x^{3}-10\times\frac{1}{2}x^{2}+24x=\frac{1}{3}x^{3}-5x^{2}+24x$.

Step5: Check by differentiation

• For $y = x^{6}$, $y^\prime=6x^{5}$.
• For $y=\frac{1}{5}x^{5}$, $y^\prime=x^{4}$.
• For $y=\frac{1}{3}x^{3}-5x^{2}+24x$, $y^\prime=x^{2}-10x + 24$.

Answer:

a. $x^{6}$ b. $\frac{1}{5}x^{5}$ c. $\frac{1}{3}x^{3}-5x^{2}+24x$