What is the integral of 5sin^3(x) ?

The integral of 5sin^3(x) is 5(-cos(x)+(cos^3(x))/3)+C

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integral of 5sin^3(x)

\int 5 sin^{3} (x) d x

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

Solution steps

\int 5 sin^{3} (x) d x

Take the constant out:

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

= 5 \cdot \int sin^{3} (x) d x

Rewrite using trig identities

= 5 \cdot \int (1 - cos^{2} (x)) sin (x) d x

Apply u-substitution

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

= 5 \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

= 5 (- \int 1 d u + \int u^{2} d u)

\int 1 d u = u

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

= 5 (- u + \frac{u ^{3}}{3})

Substitute back

u = cos (x)

= 5 (- cos (x) + \frac{cos ^{3} ( x )}{3})

Add a constant to the solution

= 5 (- cos (x) + \frac{cos ^{3} ( x )}{3}) + C

t \to 1 lim (sin (- 3 π t))

y = e^{c o s (3) x^{2}}

x^{2} + x y + 2 y^{2} = 4, (1, 1)

f (x) = 2 x^{6} + 3 x

\frac{5}{e ^{x} + e ^{- x}}

What is the integral of 5sin^3(x) ?
The integral of 5sin^3(x) is 5(-cos(x)+(cos^3(x))/3)+C