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 $\square$.

Explanation:

Step1: Recall the antiderivative formula

The antiderivative of \(\frac{1}{\sqrt{1 - x^{2}}}\) is \(\arcsin(x)+C\).

Step2: Apply the constant multiple rule

For the function \(y = \frac{3}{\sqrt{1 - x^{2}}}\), by the constant - multiple rule of integration \(\int k\cdot f(x)dx=k\int f(x)dx\) (\(k = 3\) and \(f(x)=\frac{1}{\sqrt{1 - x^{2}}}\)).
So \(\int\frac{3}{\sqrt{1 - x^{2}}}dx=3\int\frac{1}{\sqrt{1 - x^{2}}}dx\).
Since \(\int\frac{1}{\sqrt{1 - x^{2}}}dx=\arcsin(x)+C\) and \(C = 0\), we have \(\int\frac{3}{\sqrt{1 - x^{2}}}dx=3\arcsin(x)\).

Step3: Check by differentiation

Differentiate \(y = 3\arcsin(x)\) using the formula \(\frac{d}{dx}(\arcsin(x))=\frac{1}{\sqrt{1 - x^{2}}}\).
By the constant - multiple rule \(\frac{d}{dx}(3\arcsin(x))=3\cdot\frac{d}{dx}(\arcsin(x))\).
\(\frac{d}{dx}(3\arcsin(x))=\frac{3}{\sqrt{1 - x^{2}}}\)

Answer:

\(3\arcsin(x)\)