What is the integral of 5sin(t)cos(t) ?

The integral of 5sin(t)cos(t) is 5/2 sin^2(t)+C

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integral of 5sin(t)cos(t)

\int 5 sin (t) cos (t) d t

\frac{5}{2} sin^{2} (t) + C

Solution steps

\int 5 sin (t) cos (t) d t

Take the constant out:

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

= 5 \cdot \int sin (t) cos (t) d t

Apply u-substitution

= 5 \cdot \int u d u

Apply the Power Rule

= 5 \cdot \frac{u ^{2}}{2}

Substitute back

u = sin (t)

= 5 \cdot \frac{sin ^{2} ( t )}{2}

Simplify

5 \cdot \frac{sin ^{2} ( t )}{2} : \frac{5}{2} sin^{2} (t)

= \frac{5}{2} sin^{2} (t)

Add a constant to the solution

= \frac{5}{2} sin^{2} (t) + C

What is the integral of 5sin(t)cos(t) ?
The integral of 5sin(t)cos(t) is 5/2 sin^2(t)+C