What is the integral of 1/(sqrt(tan(x))) ?

The integral of 1/(sqrt(tan(x))) is

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integral of 1/(sqrt(tan(x)))

\int \frac{1}{tan ( x )} d x

- 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + C

Solution steps

\int \frac{1}{tan ( x )} d x

Rewrite using trig identities

= \int cot (x) d x

Apply Trigonometric Substitution

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

= \int - \frac{u}{1 + u ^{2}} d u

Take the constant out:

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

= - \int \frac{u}{1 + u ^{2}} d u

Apply Integral Substitution

= - \int \frac{2 v}{4 + v ^{2}} d v

Take the constant out:

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

= - 2 \cdot \int \frac{v}{4 + v ^{2}} d v

Apply u-substitution

: \int \frac{2 w ^{2}}{4 + w ^{4}} d w

= - 2 \cdot \int \frac{2 w ^{2}}{4 + w ^{4}} d w

Take the constant out:

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

= - 2 \cdot 2 \cdot \int \frac{w ^{2}}{4 + w ^{4}} d w

Take the partial fraction of

\frac{w ^{2}}{4 + w ^{4}} : - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )}

= - 2 \cdot 2 \cdot \int - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 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

= - 2 \cdot 2 (- \int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w + \int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w)

\int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1))

\int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1))

= - 2 \cdot 2 (- \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1)) + \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1)))

Substitute back

= - 2 \cdot 2 (- \frac{1}{8} (ln (2 cot (x))^{2} + 2 2 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln (2 cot (x))^{2} - 2 2 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

Simplify

- 2 \cdot 2 (- \frac{1}{8} (ln (2 cot (x))^{2} + 2 2 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln (2 cot (x))^{2} - 2 2 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) : - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

Add a constant to the solution

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + C

\int_{0}^{0.5} 300 x d x

x^{2} y^{^{''}} + 7 x y^{^{'}} + 5 y = 0, y (1) = 0, y^{^{'}} (1) = 2

\frac{\partial}{\partial x} (x \cdot y^{2})

x \to 0 lim ((1 + \frac{x}{2})^{\frac{1}{x}})

- x^{2} + 8 x, x^{2} - 4 x