find the integral from $0$ to infinity of $5x^3 + 7e^5$

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Q. find the integral from

0

to infinity of

5x^3 + 7e^5

Write Integral Function: Write down the integral that needs to be solved. $\newline$ We need to find the integral of the function $5x^3 + 7e^5$ with respect to $x$ , from $0$ to infinity.
Split Integral Parts: Split the integral into two parts. $\newline$ The integral of a sum is the sum of the integrals, so we can write: $\newline$ $\int_{0}^{\infty} (5x^3 + 7e^5) \, dx = \int_{0}^{\infty} 5x^3 \, dx + \int_{0}^{\infty} 7e^5 \, dx$ .
Evaluate First Integral: Evaluate the first integral. $\newline$ The integral of $5x^3$ with respect to $x$ is $(\frac{5}{4})x^4$ . We need to evaluate this from $0$ to infinity.
Apply Limits First Integral: Apply the limits to the first integral. $\newline$ As $x$ approaches infinity, $(\frac{5}{4})x^4$ approaches infinity. At $x = 0$ , $(\frac{5}{4})x^4$ is $0$ . Therefore, the first integral diverges to infinity.
Evaluate Second Integral: Evaluate the second integral. $\newline$ The integral of $7e^5$ with respect to $x$ is $7e^{5x}$ , since $e^5$ is a constant with respect to $x$ . We need to evaluate this from $0$ to $\newline$fty.
Apply Limits Second Integral: Apply the limits to the second integral. $\newline$ As $x$ approaches infinity, $7e^{5x}$ approaches infinity. At $x = 0$ , $7e^{5x}$ is $0$ . Therefore, the second integral also diverges to infinity.
Combine Integral Results: Combine the results of the two integrals. $\newline$ Since both integrals diverge to infinity, the integral of $5x^3 + 7e^5$ from $0$ to infinity is also infinity.