determine the following indefinite integrals. check your work by differentiation\n int\frac{12t^{8}-t}{t^{3/2…

Explanation:

Step1: Simplify the integrand

First, rewrite $\frac{12t^{8}-t}{t^{3/2}}$ as $12t^{8 - \frac{3}{2}}-t^{1-\frac{3}{2}}=12t^{\frac{16 - 3}{2}}-t^{\frac{2 - 3}{2}}=12t^{\frac{13}{2}}-t^{-\frac{1}{2}}$.

Step2: Integrate term - by - term

Using the power rule for integration $\int t^{n}dt=\frac{t^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int(12t^{\frac{13}{2}}-t^{-\frac{1}{2}})dt=12\int t^{\frac{13}{2}}dt-\int t^{-\frac{1}{2}}dt$. $12\times\frac{t^{\frac{13}{2}+1}}{\frac{13}{2}+1}-\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C$. $12\times\frac{t^{\frac{15}{2}}}{\frac{15}{2}}-\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+C$. $12\times\frac{2}{15}t^{\frac{15}{2}}-2t^{\frac{1}{2}}+C=\frac{8}{5}t^{\frac{15}{2}}-2t^{\frac{1}{2}}+C$.

Step3: Check by differentiation

Differentiate $\frac{8}{5}t^{\frac{15}{2}}-2t^{\frac{1}{2}}+C$ using the power rule for differentiation $\frac{d}{dt}(t^{n})=nt^{n - 1}$. $\frac{d}{dt}(\frac{8}{5}t^{\frac{15}{2}}-2t^{\frac{1}{2}}+C)=\frac{8}{5}\times\frac{15}{2}t^{\frac{15}{2}-1}-2\times\frac{1}{2}t^{\frac{1}{2}-1}$. $12t^{\frac{13}{2}}-t^{-\frac{1}{2}}$, which is the original integrand.

Answer:

$\frac{8}{5}t^{\frac{15}{2}}-2t^{\frac{1}{2}}+C$