Evaluate the integral: int(-5)/(36+4x^(2))dx

Given Integral Simplification: We are given the integral: int(-5)/(36+4x^(2))dx First, we can simplify the integral by factoring out the constant from the denominator: int(-5)/(4(9+x^(2)))dx Now, we can pull out the constant -5/4 from the integral: -5/4 * int(1)/(9+x^(2))dxConstant Factor Extraction: Next, we recognize that the integral has the form of the inverse tangent function, where the integral of 1/(a^2 + x^2)dx is (1/a) * arctan(x/a) + C. In our case, a^2 = 9, so a = 3. Therefore, we can rewrite the integral as: -5/4 * (1/3) * arctan(x/3) + CInverse Tangent Function Integration: Now, we can simplify the constant factor: -5/4 * (1/3) = -5/12 So the integral becomes: -5/12 * arctan(x/3) + C

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Find indefinite integrals using the substitution

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Q. Evaluate the integral:

\int \frac{-5}{36+4 x^{2}} d x

Given Integral Simplification: We are given the integral: $\newline$ $\int \frac{-5}{36+4x^{2}}\,dx$ $\newline$ First, we can simplify the integral by factoring out the constant from the denominator: $\newline$ $\int \frac{-5}{4(9+x^{2})}\,dx$ $\newline$ Now, we can pull out the constant $-\frac{5}{4}$ from the integral: $\newline$ $-\frac{5}{4} \times \int \frac{1}{9+x^{2}}\,dx$
Constant Factor Extraction: Next, we recognize that the integral has the form of the inverse tangent function, where the integral of $\frac{1}{a^2 + x^2}$ dx is $\frac{1}{a} \cdot \text{arctan}(\frac{x}{a}) + C$ . In our case, $a^2 = 9$ , so $a = 3$ . Therefore, we can rewrite the integral as: $-\frac{5}{4} \cdot \frac{1}{3} \cdot \text{arctan}(\frac{x}{3}) + C$
Inverse Tangent Function Integration: Now, we can simplify the constant factor: $\newline$ $-\frac{5}{4} \times \frac{1}{3} = -\frac{5}{12}$ $\newline$ So the integral becomes: $\newline$ $-\frac{5}{12} \times \text{arctan}(\frac{x}{3}) + C$