Integration by Substitution/Examples/Primitive of Root of 1 minus x

Example of Use of Integration by Substitution

$\ds \int \sqrt {1 - x} \rd x = -\dfrac 2 3 \paren {1 - x}^{3 / 2} + C$

Proof

\(\ds u\)

\(=\)

\(\ds \sqrt {1 - x}\)

\(\ds x\)

\(=\)

\(\ds 1 - u^2\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d x} {\d u}\)

\(=\)

\(\ds -2 u\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \sqrt {1 - x} \rd x\)

\(=\)

\(\ds \int -2 u^2 \rd u\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds -\dfrac 2 3 u^3 + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds -\dfrac 2 3 \paren {1 - x}^{3 / 2} + C\)

substituting for $u$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): change of variable
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): change of variable