What is the integral of 6sin(2t) ?

The integral of 6sin(2t) is -3cos(2t)+C

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integral of 6sin(2t)

\int 6 sin (2 t) d t

- 3 cos (2 t) + C

Solution steps

\int 6 sin (2 t) d t

Take the constant out:

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

= 6 \cdot \int sin (2 t) d t

Apply u-substitution

: \int sin (u) \frac{1}{2} d u

= 6 \cdot \int sin (u) \frac{1}{2} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int sin (u) d u

Use the common integral:

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

= 6 \cdot \frac{1}{2} (- cos (u))

Substitute back

u = 2 t

= 6 \cdot \frac{1}{2} (- cos (2 t))

Simplify

6 \cdot \frac{1}{2} (- cos (2 t)) : - 3 cos (2 t)

= - 3 cos (2 t)

Add a constant to the solution

= - 3 cos (2 t) + C

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x \to \infty lim (2 - 1)

\int (x^{2} - 9) d x

\int \frac{x ( x - 6 )}{( x - 3 ) ^{3}} d x