91Ó°ÊÓ

In the following exercises, solve for the anti derivative \(\int f\) of \(f\) with \(C=0\), then use a calculator to graph \(f\) and the anti derivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the anti derivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$ \text { [T] } \int \frac{e^{x}}{1+e^{2 x}} d x \text { over }[-6,6] $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{\sin ^{-1} t d t}{\sqrt{1-t^{2}}} $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{d t}{\sin ^{-1} t \sqrt{1-t^{2}}} $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{\tan ^{-1}(2 t)}{1+4 t^{2}} d t $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{t \tan ^{-1}\left(t^{2}\right)}{1+t^{4}} d t $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} d t $$

In the following exercises, compute the anti derivative using appropriate substitutions. $$ \int \frac{t \sec ^{-1}\left(t^{2}\right)}{t^{2} \sqrt{t^{4}-1}} d t $$

In the following exercises, use a calculator to graph the anti derivative \(\int f\) with \(C=0\) over the given interval \([a, b]\). Approximate a value of \(C,\) if possible, such that adding \(C\) to the anti derivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$ \text { [T] } \int \frac{1}{x \sqrt{x^{2}-4}} d x \text { over }[2,6] $$

In the following exercises, use a calculator to graph the anti derivative \(\int f\) with \(C=0\) over the given interval \([a, b]\). Approximate a value of \(C,\) if possible, such that adding \(C\) to the anti derivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$ [\mathrm{T}] \int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} d x \text { over }[0,2] $$

In the following exercises, use a calculator to graph the anti derivative \(\int f\) with \(C=0\) over the given interval \([a, b]\). Approximate a value of \(C,\) if possible, such that adding \(C\) to the anti derivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). $$ \text { [T] } \int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} \text { over }[-1,1] $$