Primitive of Reciprocal of Root of x squared plus a squared/Examples/16 + 9 x^2
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Example of Use of Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$
- $\ds \int \dfrac 1 {\sqrt {16 + 9 x^2} } = \dfrac 1 3 \map \ln {3 x + \sqrt {9 x^2 + 16} }$
Proof
We have from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$:
- $\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$
Let $y = 3 x$.
Then:
| \(\ds \int \dfrac {\d x} {\sqrt {16 + 9 x^2} }\) | \(=\) | \(\ds \int \dfrac {\d x} {\sqrt {16 + \paren {3 x}^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \int \dfrac {\map \d {3 x} } {\sqrt {16 + \paren {3 x}^2} }\) | Primitive of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \map \ln {3 x + \sqrt {\paren {3 x}^2 + 16} }\) | Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3 \map \ln {3 x + \sqrt {9 x^2 + 16} }\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $14$.