What is the integral from cos(x) to sin(x) of xy ?

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

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integral from cos(x) to sin(x) of xy

\int_{c o s (x)}^{s i n (x)} x y d y

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

Solution steps

\int_{c o s (x)}^{s i n (x)} x y d y

Take the constant out:

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

= x \cdot \int_{c o s (x)}^{s i n (x)} y d y

Apply the Power Rule

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

Compute the boundaries:

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

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

Simplify

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

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