30 & 35 please be accurate ### Question**30.** Evaluate the integral: \[\int (2 + x + 2x^2 + e^x) \, dx\]### ExplanationThis problem requires finding the indefinite integral of the function \(2 + x + 2x^2 + e^x\) with respect to \(x\). The expression inside the integral combines polynomial terms with the exponential function \(e^x\). Each term can be integrated separately using basic integration rules:- The integral of a constant \(a\) is \(ax + C\).- The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).- The integral of \(e^x\) is \(e^x + C\).Apply these rules to solve the integral, representing the result in a clear and simplified form. ### Integral Problem**Problem 35: Evaluate the integral**\[\int \left( \sqrt{x} + \frac{2}{\sqrt{x}} \right) \, dx\]---This problem involves finding the indefinite integral of the sum of two functions: the square root of \(x\) and the reciprocal of the square root of \(x\) multiplied by 2. **Steps for solving:**1. **Decompose the integral** into two separate integrals: \[ \int \sqrt{x} \, dx + \int \frac{2}{\sqrt{x}} \, dx \]2. **Integrate each part individually:** - For the first integral, use the power rule for integrals: \[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} + C_1 \] - For the second integral, rewrite the integrand in exponential form and apply the power rule: \[ \int 2x^{-1/2} \, dx = 2 \cdot \frac{x^{1/2}}{1/2} = 4x^{1/2} + C_2 \]3. **Combine the results:** \[ \frac{2}{3} x^{3/2} + 4x^{1/2} + C \]Where \(C\) is the constant of integration representing an indefinite integral.

Name: 30 & 35 please be accurate ### Question**30.** Evaluate the integral:
Uploaded: 2024-03-20T06:46:03.000Z
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Description: 30 & 35 please be accurate ### Question**30.** Evaluate the integral: \[\int (2 + x + 2x^2 + e^x) \, dx\]### ExplanationThis problem requires finding the indefinite integral of the function \(2 + x + 2x^2 + e^x\) with respect to \(x\). The expressio