Questions: Evaluate the integral. [ int1^9left(frac8x^2+5 xright) d x ]

Solution Steps

To evaluate the integral \(\int_{1}^{9}\left(\frac{8}{x^{2}}+5 x\right) dx\), we can split it into two separate integrals and solve each one individually. The first integral is \(\int_{1}^{9} \frac{8}{x^{2}} dx\) and the second is \(\int_{1}^{9} 5x dx\). We then sum the results of these integrals.

We need to evaluate the integral

\[ \int_{1}^{9}\left(\frac{8}{x^{2}}+5 x\right) dx. \]

This can be split into two separate integrals:

\[ \int_{1}^{9} \frac{8}{x^{2}} dx + \int_{1}^{9} 5x dx. \]

The first integral is

\[ \int_{1}^{9} \frac{8}{x^{2}} dx. \]

The antiderivative of \(\frac{8}{x^{2}}\) is \(-\frac{8}{x}\). Evaluating this from 1 to 9 gives:

\[ \left[-\frac{8}{x}\right]_{1}^{9} = -\frac{8}{9} - \left(-8\right) = -\frac{8}{9} + 8 = -\frac{8}{9} + \frac{72}{9} = \frac{64}{9}. \]

The second integral is

\[ \int_{1}^{9} 5x dx. \]

The antiderivative of \(5x\) is \(\frac{5}{2}x^{2}\). Evaluating this from 1 to 9 gives:

\[ \left[\frac{5}{2}x^{2}\right]_{1}^{9} = \frac{5}{2}(9^{2}) - \frac{5}{2}(1^{2}) = \frac{5}{2}(81) - \frac{5}{2}(1) = \frac{405}{2} - \frac{5}{2} = \frac{400}{2} = 200. \]

Now, we combine the results of the two integrals:

\[ \frac{64}{9} + 200. \]

To add these, we convert 200 to a fraction with a denominator of 9:

\[ 200 = \frac{1800}{9}. \]

Thus, we have:

\[ \frac{64}{9} + \frac{1800}{9} = \frac{1864}{9}. \]

Final Answer