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6-101.

Your new derivative formulas can be reversed to enable you to integrate new expressions. Use this idea in each of the following problems.

$\int _ { 0 } ^ { 1 } [ \frac { 1 } { 1 + x ^ { 2 } } ] d x$
Hint (a):
You should recognize the integrand. Think inverse trig!

$\int \frac { 4 } { \sqrt { 1 - y ^ { 2 } } } d y$
Hint (b):
Refer to the hint in part (a).

$\int _ { 1 } ^ { e } ( \frac { 2 } { x } ) d x$
Hint (c):
Before integrating, rewrite the integrand with a negative exponent.

$\int - \frac { 3 } { \sqrt { 1 - ( 3 t ) ^ { 2 } } } d t$
Hint (d):
Refer to the hint in part (a).
Answer (d):
There are two equivalent expressions for this integral.
It could $=\operatorname{arccos}(3t) + C$ or it could $= −\operatorname{arcsin}(3t) + C$ .
Think about it.

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