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 power - rule for antiderivatives states that if (f(x)=x^n), then (F(x)=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). Here (C = 0).

Step2: Find antiderivative of (a. f(x)=x^{\sqrt{19}})

Using the power - rule, (F(x)=\frac{x^{\sqrt{19}+1}}{\sqrt{19}+1}) since (n=\sqrt{19}).

Step3: Find antiderivative of (b. f(x)=x^{4\pi})

Using the power - rule with (n = 4\pi), (F(x)=\frac{x^{4\pi + 1}}{4\pi+1}).

Step4: Find antiderivative of (c. f(x)=x^{\sqrt{29}-1})

Using the power - rule with (n=\sqrt{29}-1), (F(x)=\frac{x^{\sqrt{29}-1 + 1}}{\sqrt{29}-1+1}=\frac{x^{\sqrt{29}}}{\sqrt{29}}).

Step5: Check by differentiation

For (a), if (F(x)=\frac{x^{\sqrt{19}+1}}{\sqrt{19}+1}), then (F^\prime(x)=\frac{(\sqrt{19}+1)x^{\sqrt{19}}}{\sqrt{19}+1}=x^{\sqrt{19}}) by the power - rule for differentiation ((x^n)^\prime=nx^{n - 1}). For (b), if (F(x)=\frac{x^{4\pi + 1}}{4\pi+1}), then (F^\prime(x)=\frac{(4\pi + 1)x^{4\pi}}{4\pi+1}=x^{4\pi}). For (c), if (F(x)=\frac{x^{\sqrt{29}}}{\sqrt{29}}), then (F^\prime(x)=\frac{\sqrt{29}x^{\sqrt{29}-1}}{\sqrt{29}}=x^{\sqrt{29}-1}).

Answer:

a. (\frac{x^{\sqrt{19}+1}}{\sqrt{19}+1}) b. (\frac{x^{4\pi + 1}}{4\pi+1}) c. (\frac{x^{\sqrt{29}}}{\sqrt{29}})