Content

A.8. Integrals with $a^{2} - u^{2}$

Usual real conditions: $a > 0$ , $∣ u ∣ \leq a$ ; if the radical or $u$ appears in the denominator, the endpoints or $u = 0$ must be excluded, as appropriate.

Formula

\int a^{2} - u^{2} d u = \frac{u}{2} a^{2} - u^{2} + \frac{a ^{2}}{2} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

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Formula

\int u^{2} a^{2} - u^{2} d u = \frac{u}{8} (2 u^{2} - a^{2}) a^{2} - u^{2} + \frac{a ^{4}}{8} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

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Formula

\int \frac{a ^{2} - u ^{2}}{u} d u = a^{2} - u^{2} - a ln \frac{a + a ^{2} - u ^{2}}{u} + C

$ln \frac{a + a ^{2} - u ^{2}}{u} + C$
Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

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Formula

\int \frac{a ^{2} - u ^{2}}{u ^{2}} d u = - \frac{1}{u} a^{2} - u^{2} - arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

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Formula

\int \frac{u ^{2} d u}{a ^{2} - u ^{2}} = - \frac{u}{2} a^{2} - u^{2} + \frac{a ^{2}}{2} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{u a ^{2} - u ^{2}} = - \frac{1}{a} ln \frac{a + a ^{2} - u ^{2}}{u} + C

$ln \frac{a + a ^{2} - u ^{2}}{u} + C$
Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{u ^{2} a ^{2} - u ^{2}} = - \frac{a ^{2} - u ^{2}}{a ^{2} u} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int (a^{2} - u^{2})^{3/2} d u = - \frac{u}{8} (2 u^{2} - 5 a^{2}) a^{2} - u^{2} + \frac{3 a ^{4}}{8} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{( a ^{2} - u ^{2} ) ^{3/2}} = \frac{u}{a ^{2} a ^{2} - u ^{2}} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Content

A.8. Integrals with $a^{2} - u^{2}$

Usual real conditions: $a > 0$ , $∣ u ∣ \leq a$ ; if the radical or $u$ appears in the denominator, the endpoints or $u = 0$ must be excluded, as appropriate.

Formula

\int a^{2} - u^{2} d u = \frac{u}{2} a^{2} - u^{2} + \frac{a ^{2}}{2} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int u^{2} a^{2} - u^{2} d u = \frac{u}{8} (2 u^{2} - a^{2}) a^{2} - u^{2} + \frac{a ^{4}}{8} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{a ^{2} - u ^{2}}{u} d u = a^{2} - u^{2} - a ln \frac{a + a ^{2} - u ^{2}}{u} + C

$ln \frac{a + a ^{2} - u ^{2}}{u} + C$
Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{a ^{2} - u ^{2}}{u ^{2}} d u = - \frac{1}{u} a^{2} - u^{2} - arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{u ^{2} d u}{a ^{2} - u ^{2}} = - \frac{u}{2} a^{2} - u^{2} + \frac{a ^{2}}{2} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{u a ^{2} - u ^{2}} = - \frac{1}{a} ln \frac{a + a ^{2} - u ^{2}}{u} + C

$ln \frac{a + a ^{2} - u ^{2}}{u} + C$
Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{u ^{2} a ^{2} - u ^{2}} = - \frac{a ^{2} - u ^{2}}{a ^{2} u} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int (a^{2} - u^{2})^{3/2} d u = - \frac{u}{8} (2 u^{2} - 5 a^{2}) a^{2} - u^{2} + \frac{3 a ^{4}}{8} arcsin \frac{u}{a} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →

Formula

\int \frac{d u}{( a ^{2} - u ^{2} ) ^{3/2}} = \frac{u}{a ^{2} a ^{2} - u ^{2}} + C

Note: In the app's source code, the last two formulas appear with $(a^{2} + u^{2})^{3/2}$ in the LaTeX; here it is corrected to $(a^{2} - u^{2})^{3/2}$ to be consistent with the theme.

View detail →