What is the integral of (1+cos(3x))^{3/2} ?

The integral of (1+cos(3x))^{3/2} is (4*2^{1/2})/3 (sin((3x)/2)-1/3 sin^3((3x)/2))+C

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integral of (1+cos(3x))^{3/2}

\int (1 + cos (3 x))^{\frac{3}{2}} d x

\frac{4 \cdot 2 ^{\frac{1}{2}}}{3} (sin (\frac{3 x}{2}) - \frac{1}{3} sin^{3} (\frac{3 x}{2})) + C

Solution steps

\int (1 + cos (3 x))^{\frac{3}{2}} d x

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{3} \cdot \int (1 + cos (u))^{\frac{3}{2}} d u

Rewrite using trig identities

= \frac{1}{3} \cdot \int 2 \cdot 2^{\frac{1}{2}} (cos^{2} (\frac{u}{2}))^{\frac{3}{2}} d u

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \int (cos^{2} (\frac{u}{2}))^{\frac{3}{2}} d u

(cos^{2} (\frac{u}{2}))^{\frac{3}{2}} = cos^{3} (\frac{u}{2}),

assuming

cos (\frac{u}{2}) \geq 0

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \int cos^{3} (\frac{u}{2}) d u

Apply u-substitution

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \int cos^{3} (v) \cdot 2 d v

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 \cdot \int cos^{3} (v) d v

Rewrite using trig identities

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

Apply u-substitution

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 \cdot \int 1 - w^{2} d w

Apply the Sum Rule:

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

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 (\int 1 d w - \int w^{2} d w)

\int 1 d w = w

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

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 (w - \frac{w ^{3}}{3})

Substitute back

= \frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 (sin (\frac{3 x}{2}) - \frac{sin ^{3} ( \frac{3 x}{2} )}{3})

Simplify

\frac{1}{3} \cdot 2 \cdot 2^{\frac{1}{2}} \cdot 2 (sin (\frac{3 x}{2}) - \frac{sin ^{3} ( \frac{3 x}{2} )}{3}) : \frac{4 \cdot 2 ^{\frac{1}{2}}}{3} (sin (\frac{3 x}{2}) - \frac{1}{3} sin^{3} (\frac{3 x}{2}))

= \frac{4 \cdot 2 ^{\frac{1}{2}}}{3} (sin (\frac{3 x}{2}) - \frac{1}{3} sin^{3} (\frac{3 x}{2}))

Add a constant to the solution

= \frac{4 \cdot 2 ^{\frac{1}{2}}}{3} (sin (\frac{3 x}{2}) - \frac{1}{3} sin^{3} (\frac{3 x}{2})) + C

What is the integral of (1+cos(3x))^{3/2} ?
The integral of (1+cos(3x))^{3/2} is (4*2^{1/2})/3 (sin((3x)/2)-1/3 sin^3((3x)/2))+C