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integral de 5/(sqrt(k-x^2))

Solución

\int \frac{5}{k - x ^{2}} d x

5 arcsin (\frac{x}{k}) + C

Pasos de solución

\int \frac{5}{k - x ^{2}} d x

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{k - x ^{2}} d x

k - x^{2} = k (1 - \frac{x ^{2}}{k})

= 5 \cdot \int \frac{1}{k ( 1 - \frac{x ^{2}}{k} )} d x

Aplicar las leyes de los exponentes:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{k} = (\frac{x}{k})^{2}

= 5 \cdot \int \frac{1}{k ( 1 - ( \frac{x}{k} ) ^{2} )} d x

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

k (1 - (\frac{x}{k})^{2}) = k (1 - (\frac{x}{k})^{2})

= 5 \cdot \int \frac{1}{k 1 - ( \frac{x}{k} ) ^{2}} d x

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \frac{1}{k} \cdot \int \frac{1}{1 - ( \frac{x}{k} ) ^{2}} d x

Aplicar integración por sustitución

: \int \frac{k}{1 - u ^{2}} d u

= 5 \cdot \frac{1}{k} \cdot \int \frac{k}{1 - u ^{2}} d u

\frac{k}{1 - u ^{2}} = k \frac{1}{1 - u ^{2}}

= 5 \cdot \frac{1}{k} \cdot \int k \frac{1}{1 - u ^{2}} d u

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \frac{1}{k} k \cdot \int \frac{1}{1 - u ^{2}} d u

Simplificar

\frac{1}{1 - u ^{2}} : \frac{1}{1 - u ^{2}}

= 5 \cdot \frac{1}{k} k \cdot \int \frac{1}{1 - u ^{2}} d u

Simplificar

= 5 \cdot \frac{1}{k} k \cdot \int \frac{1}{1 - u ^{2}} d u

Aplicar la regla de integración:

\int \frac{1}{1 - u ^{2}} d u = arcsin (u)

= 5 \cdot \frac{1}{k} k arcsin (u)

Sustituir en la ecuación

u = \frac{x}{k}

= 5 \cdot \frac{1}{k} k arcsin (\frac{x}{k})

Simplificar

5 \cdot \frac{1}{k} k arcsin (\frac{x}{k}) : 5 arcsin (\frac{x}{k})

= 5 arcsin (\frac{x}{k})

Agregar una constante a la solución

= 5 arcsin (\frac{x}{k}) + C

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