What is the integral of 8sin(t)cos(t) ?

The integral of 8sin(t)cos(t) is 4sin^2(t)+C

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integral of 8sin(t)cos(t)

\int 8 sin (t) cos (t) d t

4 sin^{2} (t) + C

Solution steps

\int 8 sin (t) cos (t) d t

Take the constant out:

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

= 8 \cdot \int sin (t) cos (t) d t

Apply u-substitution

: \int u d u

= 8 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= 8 \cdot \frac{u ^{2}}{2}

Substitute back

u = sin (t)

= 8 \cdot \frac{sin ^{2} ( t )}{2}

Simplify

8 \cdot \frac{sin ^{2} ( t )}{2} : 4 sin^{2} (t)

= 4 sin^{2} (t)

Add a constant to the solution

= 4 sin^{2} (t) + C

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What is the integral of 8sin(t)cos(t) ?
The integral of 8sin(t)cos(t) is 4sin^2(t)+C