int(-1)/(sqrt(1+x^(2)))×(1)/(1+x^(2))dx

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Simplify fractions: Simplify the integrand. The integrand is the product of two fractions, which can be combined into a single fraction by multiplying the numerators and denominators. (-1)/(sqrt(1+x^2)) × (1)/(1+x^2) = (-1)/((sqrt(1+x^2))(1+x^2))Recognize derivative for substitution: Recognize the derivative of the function inside the square root. Notice that the derivative of arctan(x) is 1/(1+x^2). This suggests a substitution where u = arctan(x), which means du = 1/(1+x^2) dx.Perform substitution: Perform the substitution. Let u = arctan(x), then du = 1/(1+x^2) dx. We can rewrite the integral in terms of u: int (-1)/(sqrt(1+x^2)) × (1)/(1+x^2) dx = int (-1)/(sqrt(1+tan^2(u))) du Since 1+tan^2(u) = sec^2(u), we have sqrt(1+tan^2(u)) = sqrt(sec^2(u)) = |sec(u)|.Use trigonometric identity: Simplify the integral using the trigonometric identity. Since sec(u) is the reciprocal of cos(u), and we know that for arctan(x), cos(u) is positive (because the range of arctan is (-π/2, π/2)), we can drop the absolute value. The integral becomes: int (-1)/sec(u) du = int (-cos(u)) duIntegrate with respect: Integrate with respect to u. The integral of -cos(u) with respect to u is -sin(u) + C, where C is the constant of integration. So, int (-cos(u)) du = -sin(u) + CSubstitute back: Substitute back in terms of x. Since u = arctan(x), we need to express sin(u) in terms of x. Using the trigonometric identity sin^2(u) + cos^2(u) = 1 and the fact that cos(u) = 1/sqrt(1+x^2), we can find sin(u). sin(u) = sqrt(1 - cos^2(u)) = sqrt(1 - (1/(1+x^2))^2)Simplify the expression for sin(u). sin(u) = sqrt(1 - (1/(1+x^2))^2) = sqrt(1 - 1/(1+x^2)^2) = sqrt((1+x^2)^2 - 1)/(1+x^2) However, this step contains a mistake in the simplification process. The correct simplification should be: sin(u) = sqrt(1 - (1/(1+x^2))^2) = sqrt((1+x^2) - 1)/(1+x^2) = x/(1+x^2)

$\int \frac{-1}{\sqrt{1+x^{2}}} \times \frac{1}{1+x^{2}} \, dx$

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$\int \frac{-1}{\sqrt{1+x^{2}}} \times \frac{1}{1+x^{2}} \, dx$