Questions: Evaluate the following indefinite integral. [ int frac1+3 t^47 t d t ] Simplify the integrand by dividing each term in the numerator by 7 t and write the resulting integral. [ int frac1+3 t^47 t d t=int square d t ] (Type an exact answer.)

$Evaluate the following indefinite integral. [ int frac1+3 t^47 t d t ] Simplify the integrand by dividing each term in the numerator by 7 t and write the resulting integral. [ int frac1+3 t^47 t d t=int square d t ] (Type an exact answer.)$

Transcript text: Evaluate the following indefinite integral. \[ \int \frac{1+3 t^{4}}{7 t} d t \] Simplify the integrand by dividing each term in the numerator by 7 t and write the resulting integral. \[ \int \frac{1+3 t^{4}}{7 t} d t=\int \square d t \] (Type an exact answer.)

Solution

Solution Steps

To evaluate the indefinite integral, first simplify the integrand by dividing each term in the numerator by $7t$. This will allow us to rewrite the integral in a simpler form, which can then be integrated term by term.

Step 1: Simplifying the Integrand

We start with the integral \[ \int \frac{1 + 3t^4}{7t} \, dt. \] To simplify the integrand, we divide each term in the numerator by $7t$: \[ \frac{1}{7t} + \frac{3t^4}{7t} = \frac{1}{7t} + \frac{3t^3}{7}. \] Thus, the integral can be rewritten as: \[ \int \left( \frac{1}{7t} + \frac{3t^3}{7} \right) dt. \]

Step 2: Integrating the Simplified Integrand

Now we integrate each term separately: \[ \int \frac{1}{7t} \, dt + \int \frac{3t^3}{7} \, dt. \] The first integral evaluates to: \[ \frac{1}{7} \log |t| + C_1, \] and the second integral evaluates to: \[ \frac{3}{7} \cdot \frac{t^4}{4} + C_2 = \frac{3t^4}{28} + C_2. \] Combining these results, we have: \[ \int \frac{1 + 3t^4}{7t} \, dt = \frac{1}{7} \log |t| + \frac{3t^4}{28} + C, \] where $C$ is the constant of integration.