Evaluate the integral and express your answer in simplest form. int(-5)/(16+16x^(2))dx Answer:

Factor out constant: Simplify the integral by factoring out the constant from the denominator. int(-5)/(16+16x^(2))dx = int(-5/16)/(1+x^(2))dxRecognize integral form: Recognize that the integral is now in the form of a constant times the integral of 1/(1+x^2), which is the arctangent function. int(-5/16)/(1+x^(2))dx = -5/16 * int(1/(1+x^(2)))dxEvaluate integral: Evaluate the integral using the fact that the integral of 1/(1+x^2) is arctan(x). -5/16 * int(1/(1+x^(2)))dx = -5/16 * arctan(x) + C

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Find indefinite integrals using the substitution

Full solution

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

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\int \frac{-5}{16+16 x^{2}} d x

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Answer:

Factor out constant: Simplify the integral by factoring out the constant from the denominator. $\int\frac{-5}{16+16x^{2}}dx = \int\frac{-5/16}{1+x^{2}}dx$
Recognize integral form: Recognize that the integral is now in the form of a constant times the integral of $1/(1+x^2)$ , which is the arctangent function. $\newline$ $\int\left(-\frac{5}{16}\right)/(1+x^{2})dx = -\frac{5}{16} \times \int\frac{1}{(1+x^{2})}dx$
Evaluate integral: Evaluate the integral using the fact that the integral of $\frac{1}{1+x^2}$ is $\arctan(x)$ . $\frac{-5}{16} \int \frac{1}{1+x^{2}}\,dx = \frac{-5}{16} \arctan(x) + C$