### Integrating Functions#### Integrate the Given Function:\[ \int \frac{\sqrt{9 - z^6}}{z^4} \, dz \]Here is a step-by-step solution to integrate the given function:1. Simplify the integrand if possible.2. Determine an appropriate method of integration (e.g., substitution, integration by parts).3. Perform the integration.4. Add the constant of integration \( + C \) after integrating.Use the provided space below to compute the integral:``` [ __________ ]+ C```Feel free to seek additional resources or consult a math textbook for guidance on solving integrals involving radicals and powers. Happy integrating! **Indefinite Integral Calculation Guide****Problem Statement:**Find the indefinite integral. You may omit absolute values from natural logarithms.**Integral to Solve:**\[ \int \frac{1}{(x+8)\sqrt{x^2 + 16x + 164}} \, dx \]**Steps to Solve:**1. **Simplify the Integrand:** - Notice the quadratic expression under the square root. - Complete the square for \(x^2 + 16x + 164\).2. **Complete the Square:** - The quadratic \(x^2 + 16x + 164\) can be rewritten as: \[ x^2 + 16x + 64 + 100 = (x + 8)^2 + 100 \] - Thus, the integral becomes: \[ \int \frac{1}{(x+8)\sqrt{(x+8)^2 + 100}} \, dx \]3. **Substitution:** - Use \( u = x + 8\), hence \( du = dx \): \[ \int \frac{1}{u \sqrt{u^2 + 100}} \, du \]4. **Simplify Using Known Integrals:** - Recognize that this is in the form for a standard integral result: \[ \int \frac{1}{u \sqrt{u^2 + a^2}} \, du = \frac{1}{a} \ln\left| \frac{u}{a + \sqrt{u^2 + a^2}} \right| + C \] - Where in this problem \( a = 10 \) (since \(100 = 10^2\)).5. **Substitute and Simplify:** - Substitute back \( u = x + 8 \): \[ \int \frac{1}{(x+8)\sqrt{(x+8)^2 + 100}} \, dx = \frac{1}{10} \ln\left|\frac{x+8}{10 + \sqrt{(x+8)^2 + 100}} \right| + C \]6. **Final Answer:** - The final integral result is: \[ \frac{1}{10} \ln\

Name: ### Integrating Functions#### Integrate the Given Function:\[ \int \f
Uploaded: 2024-03-20T06:46:42.000Z
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Description: ### Integrating Functions#### Integrate the Given Function:\[ \int \frac{\sqrt{9 - z^6}}{z^4} \, dz \]Here is a step-by-step solution to integrate the given function:1. Simplify the integrand if possible.2. Determine an appropriate method of integra