Questions: ∫ (9/(x^(4/5))) - (7x^3)/16 + (9/(2x^3)) dx

Solution Steps

To solve the given integral, we need to integrate each term separately. The integral consists of three terms:

1. \(\frac{9}{\sqrt[5]{x^{4}}}\), which can be rewritten as \(9x^{-\frac{4}{5}}\).
2. \(-\frac{7x^{3}}{16}\), which is a straightforward power function.
3. \(\frac{9}{2x^{3}}\), which can be rewritten as \(\frac{9}{2}x^{-3}\).

For each term, apply the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

The given integral is \(\int \frac{9}{\sqrt[5]{x^{4}}} - \frac{7x^{3}}{16} + \frac{9}{2x^{3}} \, dx\). We rewrite each term using exponents:

• \(\frac{9}{\sqrt[5]{x^{4}}} = 9x^{-\frac{4}{5}}\)
• \(-\frac{7x^{3}}{16} = -0.4375x^{3}\)
• \(\frac{9}{2x^{3}} = 4.5x^{-3}\)

Thus, the integral becomes \(\int \left(9x^{-\frac{4}{5}} - 0.4375x^{3} + 4.5x^{-3}\right) \, dx\).

Apply the power rule for integration \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) to each term:

• \(\int 9x^{-\frac{4}{5}} \, dx = 45.0x^{0.2}\)
• \(\int -0.4375x^{3} \, dx = -0.1094x^{4}\)
• \(\int 4.5x^{-3} \, dx = -2.25x^{-2}\)

Combine the results of the integration: \[ 45.0x^{0.2} - 0.1094x^{4} - 2.25x^{-2} + C \]

Final Answer