Questions: Find the indefinite integral and check the result by differentiation. (Remember the constant of integration.) ∫(7+8/t)^3(8/t^2) dt

Solution Steps

To solve the given integral, we can use substitution. Let \( u = 7 + \frac{8}{t} \), then find \( du \) in terms of \( dt \). Substitute \( u \) and \( du \) into the integral, solve the integral in terms of \( u \), and then substitute back to get the result in terms of \( t \). Finally, differentiate the result to verify the solution.

To solve the integral \(\int \left(7 + \frac{8}{t}\right)^3 \left(\frac{8}{t^2}\right) \, dt\), we use substitution. Let \( u = 7 + \frac{8}{t} \). Then, the derivative of \( u \) with respect to \( t \) is:

\[ \frac{du}{dt} = -\frac{8}{t^2} \]

This implies \( du = -\frac{8}{t^2} \, dt \), or equivalently, \( dt = -\frac{t^2}{8} \, du \).

Substitute \( u \) and \( du \) into the integral:

\[ \int u^3 \left(\frac{8}{t^2}\right) \, dt = \int u^3 \left(-du\right) = -\int u^3 \, du \]

The integral of \( u^3 \) is:

\[ -\int u^3 \, du = -\frac{u^4}{4} + C \]

Substitute back \( u = 7 + \frac{8}{t} \) into the expression:

\[ -\frac{(7 + \frac{8}{t})^4}{4} + C \]

Differentiate the result to verify:

\[ \frac{d}{dt}\left(-\frac{(7 + \frac{8}{t})^4}{4} + C\right) = \left(7 + \frac{8}{t}\right)^3 \left(\frac{8}{t^2}\right) \]

This confirms that the differentiation of the result matches the original integrand.

Final Answer