What is the integral of 34cos(x)cos^3(sin(x)) ?

The integral of 34cos(x)cos^3(sin(x)) is 34(sin(sin(x))-(sin^3(sin(x)))/3)+C

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integral of 34cos(x)cos^3(sin(x))

\int 34 cos (x) cos^{3} (sin (x)) d x

34 (sin (sin (x)) - \frac{sin ^{3} ( sin ( x ) )}{3}) + C

Solution steps

\int 34 cos (x) cos^{3} (sin (x)) d x

Take the constant out:

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

= 34 \cdot \int cos (x) cos^{3} (sin (x)) d x

Apply u-substitution

: \int cos^{3} (u) d u

= 34 \cdot \int cos^{3} (u) d u

Rewrite using trig identities

= 34 \cdot \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

: \int 1 - v^{2} d v

= 34 \cdot \int 1 - v^{2} d v

Apply the Sum Rule:

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

= 34 (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

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

= 34 (v - \frac{v ^{3}}{3})

Substitute back

= 34 (sin (sin (x)) - \frac{sin ^{3} ( sin ( x ) )}{3})

Add a constant to the solution

= 34 (sin (sin (x)) - \frac{sin ^{3} ( sin ( x ) )}{3}) + C

f (x) = \frac{π x}{4}

x \to - \infty lim (2^{3 x})

\frac{d v}{d t} = - 2 v, v (0) = 4

\frac{\partial}{\partial x} (3 x^{2} - 4 x y)

f (x) = (x^{2} + 2)^{2}, (0, 4)

What is the integral of 34cos(x)cos^3(sin(x)) ?
The integral of 34cos(x)cos^3(sin(x)) is 34(sin(sin(x))-(sin^3(sin(x)))/3)+C