Evaluate the integral and express your answer in simplest form. int(-5)/(25+25x^(2))dx Answer:

Factor out constant: Simplify the integral by factoring out the constant 25 from the denominator. int(-5)/(25+25x^(2))dx = int(-5/25)/(1+x^(2))dx = int(-1/5)/(1+x^(2))dxRecognize arctangent form: Recognize that the integral is now in the form of the arctangent function derivative. int(-1/5)/(1+x^(2))dx = -1/5 * int(1/(1+x^(2)))dxEvaluate using arctangent: Evaluate the integral using the arctangent function. -1/5 * int(1/(1+x^(2)))dx = -1/5 * arctan(x) + CWrite final answer: Write the final answer. Answer: -1/5 * arctan(x) + C

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Calculus

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{-5}{25+25 x^{2}} d x

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Answer:

Factor out constant: Simplify the integral by factoring out the constant $25$ from the denominator. $\newline$ $\int\frac{-5}{25+25x^{2}}dx = \int\frac{-5/25}{1+x^{2}}dx = \int\frac{-1/5}{1+x^{2}}dx$
Recognize arctangent form: Recognize that the integral is now in the form of the arctangent function derivative. $\int(-\frac{1}{5})/(1+x^{2})dx = -\frac{1}{5} \int(\frac{1}{1+x^{2}})dx$
Evaluate using arctangent: Evaluate the integral using the arctangent function. $\newline$ $-\frac{1}{5} \int \frac{1}{1+x^{2}}dx = -\frac{1}{5} \arctan(x) + C$
Write final answer: Write the final answer. $\newline$ Answer: $-\frac{1}{5} \cdot \arctan(x) + C$