What is the integral of (sqrt(a-y))/(sqrt(y)) ?

The integral of (sqrt(a-y))/(sqrt(y)) is a(arcsin(sqrt(y/a))+1/2 sin(2arcsin(sqrt(y/a))))+C

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integral of (sqrt(a-y))/(sqrt(y))

\int \frac{a - y}{y} d y

a (arcsin (\frac{y}{a}) + \frac{1}{2} sin (2 arcsin (\frac{y}{a}))) + C

Solution steps

\int \frac{a - y}{y} d y

Apply u-substitution

: \int 2 a - u^{2} d u

= \int 2 a - u^{2} d u

Take the constant out:

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

= 2 \cdot \int a - u^{2} d u

a - u^{2} = a (1 - \frac{u ^{2}}{a})

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

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{u ^{2}}{a} = (\frac{u}{a})^{2}

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

(\frac{u}{a})^{2} = \frac{u ^{2}}{a}

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

a (1 - \frac{u ^{2}}{a}) = a (1 - \frac{u ^{2}}{a})

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

Take the constant out:

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

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

Apply Trigonometric Substitution

: \int a cos^{2} (v) d v

= 2 a \cdot \int a cos^{2} (v) d v

Take the constant out:

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

= 2 a a \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 2 a a \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= 2 a a \frac{1}{2} \cdot \int 1 + cos (2 v) 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

= 2 a a \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 2 a a \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 2 a a \frac{1}{2} (arcsin (\frac{1}{a} y) + \frac{1}{2} sin (2 arcsin (\frac{1}{a} y)))

Simplify

2 a a \frac{1}{2} (arcsin (\frac{1}{a} y) + \frac{1}{2} sin (2 arcsin (\frac{1}{a} y))) : a (arcsin (\frac{y}{a}) + \frac{1}{2} sin (2 arcsin (\frac{y}{a})))

= a (arcsin (\frac{y}{a}) + \frac{1}{2} sin (2 arcsin (\frac{y}{a})))

Add a constant to the solution

= a (arcsin (\frac{y}{a}) + \frac{1}{2} sin (2 arcsin (\frac{y}{a}))) + C

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\int (8 - \frac{( 5 ^{x} )}{2}) d x

What is the integral of (sqrt(a-y))/(sqrt(y)) ?
The integral of (sqrt(a-y))/(sqrt(y)) is a(arcsin(sqrt(y/a))+1/2 sin(2arcsin(sqrt(y/a))))+C