What is the integral of 3sin^3(x) ?

The integral of 3sin^3(x) is -3cos(x)+cos^3(x)+C

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integral of 3sin^3(x)

\int 3 sin^{3} (x) d x

- 3 cos (x) + cos^{3} (x) + C

Solution steps

\int 3 sin^{3} (x) d x

Take the constant out:

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

= 3 \cdot \int sin^{3} (x) d x

Rewrite using trig identities

= 3 \cdot \int (1 - cos^{2} (x)) sin (x) d x

Apply u-substitution

: \int - 1 + u^{2} d u

= 3 \cdot \int - 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (- \int 1 d u + \int u^{2} d u)

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= 3 (- u + \frac{u ^{3}}{3})

Substitute back

u = cos (x)

= 3 (- cos (x) + \frac{cos ^{3} ( x )}{3})

Simplify

3 (- cos (x) + \frac{cos ^{3} ( x )}{3}) : - 3 cos (x) + cos^{3} (x)

= - 3 cos (x) + cos^{3} (x)

Add a constant to the solution

= - 3 cos (x) + cos^{3} (x) + C

y^{^{''}} + 16 y = - 32 sin (4 t)

3 arcsin (x^{3})

\frac{d v}{d t} = v^{2} + 5 v + 6

f (x) = x \cdot ln (x) - x

x \to 0 lim (x^{- 1} - 1)

What is the integral of 3sin^3(x) ?
The integral of 3sin^3(x) is -3cos(x)+cos^3(x)+C