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