Question

find the antiderivative for each function when c equals 0. do as many as you can mentally. check your answers by differentiation. a. $\frac{3}{\sqrt{1 - x^{2}}}$ b. $\frac{1}{4(x^{2}+1)}$ c. $\frac{1}{1 + 9x^{2}}$ a. the antiderivative of $\frac{3}{\sqrt{1 - x^{2}}}$ is $3\arcsin(x)$. b. the antiderivative of $\frac{1}{4(x^{2}+1)}$ is $\square$.

Explanation:

Step1: Recall the antiderivative formula

The antiderivative of \(\frac{1}{x^{2}+1}\) is \(\arctan(x)\).

Step2: Apply the constant multiple rule

For the function \(\frac{1}{4(x^{2}+1)}\), using the constant - multiple rule of integration \(\int k\cdot f(x)dx=k\int f(x)dx\) (where \(k = \frac{1}{4}\) and \(f(x)=\frac{1}{x^{2}+1}\)).
Since \(\int\frac{1}{x^{2}+1}dx=\arctan(x)+C\), then \(\int\frac{1}{4(x^{2}+1)}dx=\frac{1}{4}\int\frac{1}{x^{2}+1}dx\).
Substituting \(C = 0\), we get \(\frac{1}{4}\arctan(x)\).

Step3: Check by differentiation

Differentiate \(y=\frac{1}{4}\arctan(x)\) using the formula \(\frac{d}{dx}(\arctan(x))=\frac{1}{x^{2}+1}\).
By the constant - multiple rule \(\frac{dy}{dx}=\frac{1}{4}\cdot\frac{1}{x^{2}+1}\), which is the original function.

Answer:

\(\frac{1}{4}\arctan(x)\)