What is the integral from 0 to y of cos(10^6t-2y) ?

The integral from 0 to y of cos(10^6t-2y) is cos(1000000t) 1/2 sin(2y)+sin(1000000t)sin^2(y)

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integral from 0 to y of cos(10^6t-2y)

\int_{0}^{y} cos (1 0^{6} t - 2 y) d y

cos (1000000 t) \frac{1}{2} sin (2 y) + sin (1000000 t) sin^{2} (y)

Solution steps

\int_{0}^{y} cos (1 0^{6} t - 2 y) d y

Rewrite using trig identities

= \int_{0}^{y} cos (1000000 t) cos (2 y) + sin (1000000 t) sin (2 y) d y

Apply the Sum Rule:

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

= \int_{0}^{y} cos (1000000 t) cos (2 y) d y + \int_{0}^{y} sin (1000000 t) sin (2 y) d y

\int_{0}^{y} cos (1000000 t) cos (2 y) d y = cos (1000000 t) \frac{1}{2} sin (2 y)

\int_{0}^{y} sin (1000000 t) sin (2 y) d y = sin (1000000 t) sin^{2} (y)

= cos (1000000 t) \frac{1}{2} sin (2 y) + sin (1000000 t) sin^{2} (y)

\int_{1}^{3} (1 + 2 x) d x

\int_{1}^{x} t d t

\int_{0}^{x^{2}} cos (t^{4}) d t

\int_{0}^{1} [e^{x} - x] d x

\int_{0}^{\frac{π}{2}} sin (\frac{x}{2}) d x

What is the integral from 0 to y of cos(10^6t-2y) ?
The integral from 0 to y of cos(10^6t-2y) is cos(1000000t) 1/2 sin(2y)+sin(1000000t)sin^2(y)