What is the integral of (x^2cot^2(x^3))/(2csc(x^3)) ?

The integral of (x^2cot^2(x^3))/(2csc(x^3)) is 1/6 (ln|tan((x^3)/2)|+cos(x^3))+C

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integral of (x^2cot^2(x^3))/(2csc(x^3))

\int \frac{x ^{2} cot ^{2} ( x ^{3} )}{2 csc ( x ^{3} )} d x

\frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3})) + C

Solution steps

\int \frac{x ^{2} cot ^{2} ( x ^{3} )}{2 csc ( x ^{3} )} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{x ^{2} cot ^{2} ( x ^{3} )}{csc ( x ^{3} )} d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

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

Apply u-substitution

: \int \frac{u cos ( u ^{\frac{3}{2}} ) cot ( u ^{\frac{3}{2}} )}{2 x} d u

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

Take the constant out:

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

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

Rewrite using trig identities

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

Expand

\frac{1 - sin ^{2} ( u )}{sin ( u )} : \frac{1}{sin ( u )} - sin (u)

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int \frac{1}{sin ( u )} - sin (u) 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

= \frac{1}{2} \cdot \frac{1}{3} (\int \frac{1}{sin ( u )} d u - \int sin (u) d u)

\int \frac{1}{sin ( u )} d u = ln tan (\frac{u}{2})

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

= \frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{u}{2}) - (- cos (u)))

Substitute back

u = x^{3}

= \frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{x ^{3}}{2}) - (- cos (x^{3})))

Simplify

\frac{1}{2} \cdot \frac{1}{3} (ln tan (\frac{x ^{3}}{2}) - (- cos (x^{3}))) : \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3}))

= \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3}))

Add a constant to the solution

= \frac{1}{6} (ln tan (\frac{x ^{3}}{2}) + cos (x^{3})) + C

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What is the integral of (x^2cot^2(x^3))/(2csc(x^3)) ?
The integral of (x^2cot^2(x^3))/(2csc(x^3)) is 1/6 (ln|tan((x^3)/2)|+cos(x^3))+C