Substituting for roots and Euler‘s substituti
Task number: 3113
Convert the following integrals to integrals of a rational function and think about which partial fractions you get (which denominators). You no longer have to calculate the decomposition into partial fractions and the resulting primitive function.
Variant 1
\( \int{\sqrt{\frac{1-x}{1+x}}} \, dx \)
Variant 2
\( \int{\frac1x \root 3 \of {\frac{1-x}{1+x}}} \, dx \)
Variant 3
\( \int\frac{x}{\sqrt{x+1} + \root 3 \of {x + 1}} \, dx \)
Variant 4
\( \int \frac{1}{1 + \sqrt{x+1}} \, dx \)
Variant 5
\( \int\frac{1}{\sqrt{x^2 - 1}} \, dx \)
Variant 6
\( \int\frac{1}{\sqrt{x^2 + 1}} \, dx \)
Variant 7
\( \int\frac{1}{\sqrt{(x^2 - 1)^3}} \, dx \)
Variant 8
\( \int \frac{1}{1 + \sqrt{x^2 + 2x + 2}} \, dx \)
Variant 9
\( \int\sqrt{x^2 - 2x - 1} \, dx \)
Variant 10
\( \int \frac{x}{1 + \sqrt{-x^2 + 7x - 12}} \, dx \)
