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integral de (16)/(64+(2pix)^2)

Solución

\int \frac{16}{64 + ( 2 π x ) ^{2}} d x

Solución

0.31830 \dots arctan (0.78539 \dots x) + C

Pasos de solución

\int \frac{16}{64 + ( 2 π x ) ^{2}} d x

Sacar la constante:

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

= 16 \cdot \int \frac{1}{64 + ( 2 π x ) ^{2}} d x

Simplificar

(2 π x)^{2} : 4 \cdot 9.86960 \dots x^{2}

= 16 \cdot \int \frac{1}{64 + 4 \cdot 9.86960 \dots x ^{2}} d x

Simplificar

= 16 \cdot \int \frac{1}{64 + 39.47841 \dots x ^{2}} d x

Aplicar integración por sustitución

= 16 \cdot \int \frac{50.26548 \dots}{2526.61872 \dots + 2526.61872 \dots u ^{2}} d u

Sacar la constante:

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

= 16 \cdot 50.26548 \dots \cdot \int \frac{1}{2526.61872 \dots + 2526.61872 \dots u ^{2}} d u

2526.61872 \dots + 2526.61872 \dots u^{2} = 2526.61872 \dots (1 + \frac{2526.61872 \dots u ^{2}}{2526.61872 \dots})

= 16 \cdot 50.26548 \dots \cdot \int \frac{1}{2526.61872 \dots ( 1 + \frac{2526.61872 \dots u ^{2}}{2526.61872 \dots} )} d u

Sacar la constante:

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

= 16 \cdot 50.26548 \dots \cdot \frac{1}{2526.61872 \dots} \cdot \int \frac{1}{1 + \frac{2526.61872 \dots u ^{2}}{2526.61872 \dots}} d u

Simplificar

= 16 \cdot 50.26548 \dots \cdot 0.00039 \dots \cdot \int \frac{1}{1 + \frac{2526.61872 \dots u ^{2}}{2526.61872 \dots}} d u

Eliminar los terminos comunes:

2526.61872 \dots

= 16 \cdot 50.26548 \dots \cdot 0.00039 \dots \cdot \int \frac{1}{1 + u ^{2}} d u

Aplicar la regla de integración:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 16 \cdot 50.26548 \dots \cdot 0.00039 \dots arctan (u)

Sustituir en la ecuación

u = \frac{6.28318 \dots}{8} x

= 16 \cdot 50.26548 \dots \cdot 0.00039 \dots arctan (\frac{6.28318 \dots}{8} x)

Simplificar

= 0.31830 \dots arctan (0.78539 \dots x)

Agregar una constante a la solución

= 0.31830 \dots arctan (0.78539 \dots x) + C

Gráfica

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