What is the integral of (16x^2)/((sqrt(2x+5))^5) ?

The integral of (16x^2)/((sqrt(2x+5))^5) is 2((20)/(sqrt(2x+5))+2sqrt(2x+5)-(50)/(3(2x+5)^{3/2)})+C

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integral of (16x^2)/((sqrt(2x+5))^5)

\int \frac{16 x ^{2}}{( 2 x + 5 ) ^{5}} d x

2 (\frac{20}{2 x + 5} + 2 2 x + 5 - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}}) + C

Solution steps

\int \frac{16 x ^{2}}{( 2 x + 5 ) ^{5}} d x

Take the constant out:

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

= 16 \cdot \int \frac{x ^{2}}{( 2 x + 5 ) ^{5}} d x

Apply u-substitution

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

= 16 \cdot \int \frac{( u - 5 ) ^{2}}{8 u ^{\frac{5}{2}}} d u

Take the constant out:

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

= 16 \cdot \frac{1}{8} \cdot \int \frac{( u - 5 ) ^{2}}{u ^{\frac{5}{2}}} d u

Expand

\frac{( u - 5 ) ^{2}}{u ^{\frac{5}{2}}} : \frac{1}{u ^{\frac{1}{2}}} - \frac{10}{u ^{\frac{3}{2}}} + \frac{25}{u ^{\frac{5}{2}}}

= 16 \cdot \frac{1}{8} \cdot \int \frac{1}{u ^{\frac{1}{2}}} - \frac{10}{u ^{\frac{3}{2}}} + \frac{25}{u ^{\frac{5}{2}}} 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

= 16 \cdot \frac{1}{8} (\int \frac{1}{u ^{\frac{1}{2}}} d u - \int \frac{10}{u ^{\frac{3}{2}}} d u + \int \frac{25}{u ^{\frac{5}{2}}} d u)

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

\int \frac{10}{u ^{\frac{3}{2}}} d u = - \frac{20}{u}

\int \frac{25}{u ^{\frac{5}{2}}} d u = - \frac{50}{3 u ^{\frac{3}{2}}}

= 16 \cdot \frac{1}{8} (2 u^{\frac{1}{2}} - (- \frac{20}{u}) - \frac{50}{3 u ^{\frac{3}{2}}})

Substitute back

u = 2 x + 5

= 16 \cdot \frac{1}{8} (2 (2 x + 5)^{\frac{1}{2}} - (- \frac{20}{2 x + 5}) - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}})

Simplify

16 \cdot \frac{1}{8} (2 (2 x + 5)^{\frac{1}{2}} - (- \frac{20}{2 x + 5}) - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}}) : 2 (\frac{20}{2 x + 5} + 2 2 x + 5 - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}})

= 2 (\frac{20}{2 x + 5} + 2 2 x + 5 - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}})

Add a constant to the solution

= 2 (\frac{20}{2 x + 5} + 2 2 x + 5 - \frac{50}{3 ( 2 x + 5 ) ^{\frac{3}{2}}}) + C

\int \frac{x + 4}{x ^{2} + 4 x + 4} d x

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

\frac{\partial}{\partial y} (ln (x + y))

\frac{d y}{d x} = x x - 24

x^{2} + y^{2} = 29, (2, - 5)

What is the integral of (16x^2)/((sqrt(2x+5))^5) ?
The integral of (16x^2)/((sqrt(2x+5))^5) is 2((20)/(sqrt(2x+5))+2sqrt(2x+5)-(50)/(3(2x+5)^{3/2)})+C