exact decimal. \( \int\left(8-7 x^{4}+\frac{3}{x^{16}}+18 \sqrt{x^{7}}\right) d x \)

To find the exact decimal of the integral \( \int\left(8-7 x^{4}+\frac{3}{x^{16}}+18 \sqrt{x^{7}}\right) d x \), we’ll first integrate each term separately. 1. The integral of \( 8 \) is \( 8x \). 2. The integral of \( -7x^4 \) is \( -\frac{7}{5} x^5 \). 3. The integral of \( \frac{3}{x^{16}} \) can be rewritten as \( 3x^{-16} \) leading to \( -\frac{3}{15} x^{-15} = -\frac{1}{5} x^{-15} \). 4. The integral of \( 18\sqrt{x^7} \) is \( 18x^{\frac{7}{2}} \times \frac{2}{9} = 4 x^{\frac{9}{2}} \). Putting it all together, we have: \[ \int\left(8-7x^{4}+\frac{3}{x^{16}}+18\sqrt{x^{7}}\right) d x = 8x - \frac{7}{5} x^5 - \frac{1}{5} x^{-15} + 4 x^{\frac{9}{2}} + C \] To express this in exact decimal form, you can evaluate at specific \( x \) values or calculate constants if needed, but the expression results in a polynomial and a rational function representing the exact integral. And there you have it—the integral in all its glory! Don't forget to add the constant of integration \( C \) to account for those sneaky constant terms. Keep practicing those integrals; they might just pop up in unexpected places!