Question

find the most general antiderivative or indefinite integral.\\(\int \frac{t \sqrt7{t} + \sqrt5{t^2}}{t^2} \\, dt\\)\\(\int \frac{t \sqrt7{t} + \sqrt5{t^2}}{t^2} \\, dt = \square\\)

Explanation:

Step1: Simplify the integrand

First, rewrite the radicals as exponents. Recall that \(\sqrt[n]{t^m}=t^{\frac{m}{n}}\). So, \(\sqrt[7]{t}=t^{\frac{1}{7}}\) and \(\sqrt[5]{t^2}=t^{\frac{2}{5}}\). Then the numerator becomes \(t\cdot t^{\frac{1}{7}}+t^{\frac{2}{5}} = t^{1 + \frac{1}{7}}+t^{\frac{2}{5}}=t^{\frac{8}{7}}+t^{\frac{2}{5}}\) (using the rule \(a^m\cdot a^n=a^{m + n}\)). Now, divide each term by \(t^2\): \(\frac{t^{\frac{8}{7}}}{t^2}+\frac{t^{\frac{2}{5}}}{t^2}=t^{\frac{8}{7}-2}+t^{\frac{2}{5}-2}\) (using the rule \(\frac{a^m}{a^n}=a^{m - n}\)). Calculate the exponents: \(\frac{8}{7}-2=\frac{8 - 14}{7}=-\frac{6}{7}\) and \(\frac{2}{5}-2=\frac{2 - 10}{5}=-\frac{8}{5}\). So the integrand simplifies to \(t^{-\frac{6}{7}}+t^{-\frac{8}{5}}\).

Step2: Integrate term - by - term

Use the power rule for integration \(\int t^n dt=\frac{t^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).

• For the first term \(t^{-\frac{6}{7}}\):

\(n =-\frac{6}{7}\), so \(n + 1=-\frac{6}{7}+1=\frac{1}{7}\). Then \(\int t^{-\frac{6}{7}}dt=\frac{t^{\frac{1}{7}}}{\frac{1}{7}}+C_1 = 7t^{\frac{1}{7}}+C_1\).

• For the second term \(t^{-\frac{8}{5}}\):

\(n=-\frac{8}{5}\), so \(n + 1=-\frac{8}{5}+1=-\frac{3}{5}\). Then \(\int t^{-\frac{8}{5}}dt=\frac{t^{-\frac{3}{5}}}{-\frac{3}{5}}+C_2=-\frac{5}{3}t^{-\frac{3}{5}}+C_2\).

Step3: Combine the results

Combine the two integrals and the constants \(C_1\) and \(C_2\) into a single constant \(C\). So \(\int\frac{t\sqrt[7]{t}+\sqrt[5]{t^2}}{t^2}dt=7t^{\frac{1}{7}}-\frac{5}{3}t^{-\frac{3}{5}}+C\). We can rewrite \(t^{\frac{1}{7}}\) as \(\sqrt[7]{t}\) and \(t^{-\frac{3}{5}}\) as \(\frac{1}{t^{\frac{3}{5}}}=\frac{1}{\sqrt[5]{t^3}}\), so the integral is \(7\sqrt[7]{t}-\frac{5}{3\sqrt[5]{t^3}}+C\) (or in exponent form \(7t^{\frac{1}{7}}-\frac{5}{3}t^{-\frac{3}{5}}+C\)).

Answer:

\(7t^{\frac{1}{7}}-\frac{5}{3}t^{-\frac{3}{5}}+C\) (or \(7\sqrt[7]{t}-\frac{5}{3\sqrt[5]{t^3}}+C\))