integral de 5/(x^2)sin^2(5/x)

Solución

\int \frac{5}{x ^{2}} sin^{2} (\frac{5}{x}) d x

\frac{5}{2} (- \frac{1}{x} + \frac{1}{10} sin (\frac{10}{x})) + C

Pasos de solución

\int \frac{5}{x ^{2}} sin^{2} (\frac{5}{x}) d x

Sacar la constante:

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

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

Simplificar

\frac{1}{x ^{2}} sin^{2} (\frac{5}{x}) : \frac{sin ^{2} ( \frac{5}{x} )}{x ^{2}}

= 5 \cdot \int \frac{sin ^{2} ( \frac{5}{x} )}{x ^{2}} d x

Re-escribir usando identidades trigonométricas

= 5 \cdot \int \frac{1 - cos ( \frac{10}{x} )}{2 x ^{2}} d x

Sacar la constante:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{1 - cos ( \frac{10}{x} )}{x ^{2}} d x

Expandir

\frac{1 - cos ( \frac{10}{x} )}{x ^{2}} : \frac{1}{x ^{2}} - \frac{cos ( \frac{10}{x} )}{x ^{2}}

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{x ^{2}} - \frac{cos ( \frac{10}{x} )}{x ^{2}} d x

Aplicar la regla de la suma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 \cdot \frac{1}{2} (\int \frac{1}{x ^{2}} d x - \int \frac{cos ( \frac{10}{x} )}{x ^{2}} d x)

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

\int \frac{cos ( \frac{10}{x} )}{x ^{2}} d x = - \frac{1}{10} sin (\frac{10}{x})

= 5 \cdot \frac{1}{2} (- \frac{1}{x} - (- \frac{1}{10} sin (\frac{10}{x})))

Simplificar

5 \cdot \frac{1}{2} (- \frac{1}{x} - (- \frac{1}{10} sin (\frac{10}{x}))) : \frac{5}{2} (- \frac{1}{x} + \frac{1}{10} sin (\frac{10}{x}))

= \frac{5}{2} (- \frac{1}{x} + \frac{1}{10} sin (\frac{10}{x}))

Agregar una constante a la solución

= \frac{5}{2} (- \frac{1}{x} + \frac{1}{10} sin (\frac{10}{x})) + C

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