What is the integral from 2x to x^2 of cos(t)sin(t) ?

The integral from 2x to x^2 of cos(t)sin(t) is (sin^2(x^2)-sin^2(2x))/2

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integral from 2x to x^2 of cos(t)sin(t)

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

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

Solution steps

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

Apply u-substitution

: \int_{s i n (2 x)}^{s i n (x^{2})} u d u

= \int_{s i n (2 x)}^{s i n (x^{2})} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{s i n (2 x)}^{s i n (x^{2})}

= [\frac{u ^{2}}{2}]_{s i n (2 x)}^{s i n (x^{2})}

Compute the boundaries:

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

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

Simplify

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

\int_{0}^{1 - x} \frac{( 1 - x - y ) ^{2}}{2} d y

\int_{0}^{2} x^{2} + x + 1 d x

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

\int_{0}^{1} 3 x (x^{2} + 1)^{3} d x

\int_{0}^{x} (x - t) f (t) d t

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