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romanna [79]
3 years ago
14

Evaluate the integral

Mathematics
1 answer:
Kruka [31]3 years ago
3 0

Hello, please consider the following.

\displaystyle \begin{aligned} \int\limits^x {5sin(5t)sin(t)} \, dt &= -\int\limits^x {5sin(5t)} \, d(cos(t))\\ \\&=-[5sin(5t)cos(t)]+ \int\limits^x {25cos(5t)cos(t)} \, dt\\\\&=-5sin(5x)cos(x)+ \int\limits^x {25cos(5t)} \, d(sin(t))\\ \\&=-5sin(5x)cos(x)+[25cos(5t)sin(t)]+ \int\limits^x {25sin(5t)sin(t)} \, dt\\\\&=-5sin(5x)cos(x)+25cos(5x)sin(x)+ \int\limits^x {(25*5)sin(5t)sin(t)} \, dt\end{aligned}

And we can recognise the same integral, so.

\displaystyle (25-1)\int\limits^x {5sin(5t)sin(t)} \, dt= +5sin(5x)cos(x)-25cos(5x)sin(x)

And then,

\displaystyle \Large \boxed{\sf \bf\int\limits^x {5sin(5t)sin(t)} \, dt=\dfrac{5sin(5x)cos(x)-25cos(5x)sin(x)}{24}+C}

Thanks

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