What is the integral of (11)/(x^3)tan(9/(x^2)) ?

The integral of (11)/(x^3)tan(9/(x^2)) is 11/18 ln|cos(9/(x^2))|+C

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integral of (11)/(x^3)tan(9/(x^2))

\int \frac{11}{x ^{3}} tan (\frac{9}{x ^{2}}) d x

\frac{11}{18} ln cos (\frac{9}{x ^{2}}) + C

Solution steps

\int \frac{11}{x ^{3}} tan (\frac{9}{x ^{2}}) d x

Take the constant out:

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

= 11 \cdot \int \frac{1}{x ^{3}} tan (\frac{9}{x ^{2}}) d x

Apply u-substitution

: \int \frac{tan ( \frac{9}{u ^{\frac{2}{3}}} )}{3 u ^{\frac{5}{3}}} d u

= 11 \cdot \int \frac{tan ( \frac{9}{u ^{\frac{2}{3}}} )}{3 u ^{\frac{5}{3}}} d u

Take the constant out:

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

= 11 \cdot \frac{1}{3} \cdot \int \frac{tan ( \frac{9}{u ^{\frac{2}{3}}} )}{u ^{\frac{5}{3}}} d u

Apply u-substitution

: \int \frac{3 tan ( \frac{9}{v} )}{2 v ^{2}} d v

= 11 \cdot \frac{1}{3} \cdot \int \frac{3 tan ( \frac{9}{v} )}{2 v ^{2}} d v

Take the constant out:

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

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int \frac{tan ( \frac{9}{v} )}{v ^{2}} d v

Apply u-substitution

: \int - \frac{tan ( w )}{9} d w

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int - \frac{tan ( w )}{9} d w

Take the constant out:

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

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} \cdot \int tan (w) d w)

Rewrite using trig identities

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} \cdot \int \frac{sin ( w )}{cos ( w )} d w)

Apply u-substitution

: \int - \frac{1}{t} d t

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} \cdot \int - \frac{1}{t} d t)

Take the constant out:

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

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} (- \int \frac{1}{t} d t))

Use the common integral:

\int \frac{1}{t} d t = ln (∣ t ∣)

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} (- ln ∣ t ∣))

Substitute back

= 11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} (- ln cos (\frac{9}{( x ^{3} ) ^{\frac{2}{3}}})))

Simplify

11 \cdot \frac{1}{3} \cdot \frac{3}{2} (- \frac{1}{9} (- ln cos (\frac{9}{( x ^{3} ) ^{\frac{2}{3}}}))) : \frac{11}{18} ln cos (\frac{9}{x ^{2}})

= \frac{11}{18} ln cos (\frac{9}{x ^{2}})

Add a constant to the solution

= \frac{11}{18} ln cos (\frac{9}{x ^{2}}) + C

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What is the integral of (11)/(x^3)tan(9/(x^2)) ?
The integral of (11)/(x^3)tan(9/(x^2)) is 11/18 ln|cos(9/(x^2))|+C