Evaluate the integral and express your answer in simplest form. int(2)/(25+25x^(2))dx Answer:

Factor out common factor: Simplify the integrand by factoring out the common factor of 25 in the denominator. int(2)/(25+25x^(2))dx = int(2/25)/(1+x^(2))dx = (2/25) * int(1)/(1+x^(2))dxRecognize standard integral: Recognize that the integral of 1/(1+x^2) is a standard integral that corresponds to the arctangent function. int(1)/(1+x^(2))dx = arctan(x) + CMultiply by constant factor: Multiply the result of the integral by the constant factor that was factored out in Step 1. (2/25) * int(1)/(1+x^(2))dx = (2/25) * arctan(x) + CWrite final answer: Write the final answer, which is the simplest form of the evaluated integral. Answer: (2/25) * arctan(x) + C

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Find indefinite integrals using the substitution

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Q. Evaluate the integral and express your answer in simplest form.

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\int \frac{2}{25+25 x^{2}} d x

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Answer:

Factor out common factor: Simplify the integrand by factoring out the common factor of $25$ in the denominator. $\int\frac{2}{25+25x^{2}}dx = \int\frac{\frac{2}{25}}{1+x^{2}}dx$ = $\frac{2}{25}$ * $\int\frac{1}{1+x^{2}}dx$
Recognize standard integral: Recognize that the integral of $\frac{1}{1+x^2}$ is a standard integral that corresponds to the arctangent function. $\newline$ $\int \frac{1}{1+x^{2}}dx = \arctan(x) + C$
Multiply by constant factor: Multiply the result of the integral by the constant factor that was factored out in Step $1$ . $\newline$ $(\frac{2}{25}) \times \int \frac{1}{1+x^{2}}dx = (\frac{2}{25}) \times \arctan(x) + C$
Write final answer: Write the final answer, which is the simplest form of the evaluated integral. $\newline$ Answer: $(\frac{2}{25}) \cdot \arctan(x) + C$