Evaluate the integral and express your answer in simplest form. int(-2)/(sqrt(1-25x^(2)))dx Answer:

Recognize Integral Form: We are given the integral: int(-2)/(sqrt(1-25x^(2)))dx To solve this integral, we recognize that it resembles the derivative of the arcsine function, where the integral of 1/sqrt(1-u^2)du is arcsin(u) + C. We will use a substitution to transform the integral into this form. Let u = 5x, then du/dx = 5 and dx = du/5.Perform Substitution: Substitute u = 5x and dx = du/5 into the integral: int(-2)/(sqrt(1-25x^(2)))dx = int(-2)/(sqrt(1-u^2))(du/5) Simplify the integral: = -2/5 * int(1/sqrt(1-u^2))du Now the integral is in the form of the derivative of arcsin(u).Simplify Integral: Evaluate the integral: -2/5 * int(1/sqrt(1-u^2))du = -2/5 * arcsin(u) + C Replace u with 5x to return to the original variable: = -2/5 * arcsin(5x) + C This is the simplest form of the antiderivative.

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Evaluate the integral and express your answer in simplest form.

int(-2)/(sqrt(1-25x^(2)))dx
Answer:

Evaluate the integral and express your answer in simplest form. $\newline$ $\int \frac{-2}{\sqrt{1-25 x^{2}}} d x$ $\newline$ Answer:

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

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\int \frac{-2}{\sqrt{1-25 x^{2}}} d x

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Answer:

Recognize Integral Form: We are given the integral: $\newline$ $\int \frac{-2}{\sqrt{1-25x^{2}}}dx$ $\newline$ To solve this integral, we recognize that it resembles the derivative of the arcsine function, where the integral of $\frac{1}{\sqrt{1-u^2}}du$ is $\text{arcsin}(u) + C$ . We will use a substitution to transform the integral into this form. $\newline$ Let $u = 5x$ , then $\frac{du}{dx} = 5$ and $dx = \frac{du}{5}$ .
Perform Substitution: Substitute $u = 5x$ and $dx = \frac{du}{5}$ into the integral: $\newline$ $\int \frac{-2}{\sqrt{1-25x^{2}}}dx = \int \frac{-2}{\sqrt{1-u^2}}\left(\frac{du}{5}\right)$ $\newline$ Simplify the integral: $\newline$ $= \frac{-2}{5} \times \int \frac{1}{\sqrt{1-u^2}}du$ $\newline$ Now the integral is in the form of the derivative of $\arcsin(u)$ .
Simplify Integral: Evaluate the integral: $\newline$ $-\frac{2}{5} \int \frac{1}{\sqrt{1-u^2}}du = -\frac{2}{5} \arcsin(u) + C$ $\newline$ Replace $u$ with $5x$ to return to the original variable: $\newline$ $= -\frac{2}{5} \arcsin(5x) + C$ $\newline$ This is the simplest form of the antiderivative.