What is the integral from pi/4 to x of cos(2t) ?

The integral from pi/4 to x of cos(2t) is 1/2 (sin(2x)-1)

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integral from pi/4 to x of cos(2t)

\int_{\frac{π}{4}}^{x} cos (2 t) d t

\frac{1}{2} (sin (2 x) - 1)

Solution steps

\int_{\frac{π}{4}}^{x} cos (2 t) d t

Apply u-substitution

: \int_{\frac{π}{2}}^{2 x} cos (u) \frac{1}{2} d u

= \int_{\frac{π}{2}}^{2 x} cos (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{\frac{π}{2}}^{2 x} cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} [sin (u)]_{\frac{π}{2}}^{2 x}

Compute the boundaries:

sin (2 x) - 1

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

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\int_{1}^{4} 25 - x^{2} d x

\int_{1}^{5} ln (t^{2} + 5 t + 1) d t

What is the integral from pi/4 to x of cos(2t) ?
The integral from pi/4 to x of cos(2t) is 1/2 (sin(2x)-1)