Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int \frac{t^{2}+2}{t^{2}} d t $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{t^2+2}{t^2} dt\) is \( t - 2t^{-1} + C\)

Step 1: Simplification of the Integral Term

Firstly, rewrite the integral to simplify it. The integral term \(\frac{t^2+2}{t^2}\) can be simplified as \(\frac{t^2}{t^2} + \frac{2}{t^2}\), which is \(1 + \frac{2}{t^2}\). So, the given integral \(\int \frac{t^2+2}{t^2} dt\) becomes \(\int (1 + \frac{2}{t^2}) dt\)

Step 2: Integration

Now integrate each term separately. The integral of 1 w.r.t. \( t\) is \( t\), and the integral of \(\frac{2}{t^2}\) w.r.t. \( t\) is \( -2t^{-1}\). So, the indefinite integral of the given function becomes \( t - 2t^{-1} + C\), where \( C\) is the constant of integration.

Step 3: Checking the Result by Differentiation

Differentiate the result of the indefinite integral, \( t - 2t^{-1} + C\), w.r.t. \( t\). The derivative of \( t\) is \( 1\), and the derivative of \( -2t^{-1}\) is \( \frac{2}{t^2}\). Therefore, the derivative of the indefinite integral is \( 1 + \frac{2}{t^2}\) which is the same as the integrand of the original function. This confirms that the answer is correct.