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

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

3 0

2 years ago

Which spreadsheet would you use to compute the first seven terms of the arithmetic sequence: an = –6(n – 1) 22?

Greeley [361]

Here, just substitute the value of n = 7
an = -6(n-1)
an = -6(7-1)
an = -6(6)
an = -36

In short, Your Final Answer would be -36

Hope this helps!

6 0

3 years ago

On Monday it rained two and one-half inches. On Tuesday it rained three and three-fourths inches. How much did it rain in the tw

charle [14.2K]

Answer:

it rained 6 1/4 or 6.25 inches in two days

3 0

2 years ago