91Ó°ÊÓ

Explain the relationship \(-\cos ^{-1} t+C=\int \frac{d t}{\sqrt{1-t^{2}}}=\sin ^{-1} t+C .\) Is it true, in general, that \(\cos ^{-1} t=-\sin ^{-1} t ?\)

Explain the relationship \(\sec ^{-1} t+C=\int \frac{d t}{|t| \sqrt{t^{2}-1}}=-\csc ^{-1} t+C .\) Is it true, in general, that \(\sec ^{-1} t=-\csc ^{-1} t ?\)

Explain what is wrong with the following integral: $$\int_{1}^{2} \frac{d t}{\sqrt{1-t^{2}}}$$

Explain what is wrong with the following integral: $$\int_{-1}^{1} \frac{d t}{|t| \sqrt{t^{2}-1}}$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{1}{\sqrt{9-x^{2}}} d x \text { over }[-3,3]$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b] .\) Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t .\) $$[\mathbf{T}] \int \frac{9}{9+x^{2}} d x \text { over } [-6,6]$$

In the following exercises, solve for the antiderivative \(\int f\) of \(f\) with \(C=0,\) then use a calculator to graph \(f\) and the antiderivative over the given interval \([a, b]\). Identify a value of \(C\) such that adding \(C\) to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{\cos x}{4+\sin ^{2} x} d x\) over [-6,6]

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

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

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