What is the integral of (4sqrt(x))/(sqrt(4x+1)) ?

The integral of (4sqrt(x))/(sqrt(4x+1)) is -1/2 ln|2sqrt(x)+sqrt(4x+1)|+sqrt(x)sqrt(4x+1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4sqrt(x))/(sqrt(4x+1))

\int \frac{4 x}{4 x + 1} d x

- \frac{1}{2} ln 2 x + 4 x + 1 + x 4 x + 1 + C

Solution steps

\int \frac{4 x}{4 x + 1} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{4 x + 1} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{4 u ^{2} + 1} d u

= 4 \cdot \int \frac{2 u ^{2}}{4 u ^{2} + 1} d u

Take the constant out:

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

= 4 \cdot 2 \cdot \int \frac{u ^{2}}{4 u ^{2} + 1} d u

Apply Trigonometric Substitution

: \int \frac{tan ^{2} ( v ) sec ( v )}{8} d v

= 4 \cdot 2 \cdot \int \frac{tan ^{2} ( v ) sec ( v )}{8} d v

Take the constant out:

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

= 4 \cdot 2 \cdot \frac{1}{8} \cdot \int tan^{2} (v) sec (v) d v

Rewrite using trig identities

= 4 \cdot 2 \cdot \frac{1}{8} \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

Expand

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 4 \cdot 2 \cdot \frac{1}{8} \cdot \int - sec (v) + sec^{3} (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

= 4 \cdot 2 \cdot \frac{1}{8} (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 4 \cdot 2 \cdot \frac{1}{8} (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= 4 \cdot 2 \cdot \frac{1}{8} (- ln tan (arctan (2 x)) + sec (arctan (2 x)) + \frac{1}{2} sec (arctan (2 x)) tan (arctan (2 x)) + \frac{1}{2} ln tan (arctan (2 x)) + sec (arctan (2 x)))

Simplify

4 \cdot 2 \cdot \frac{1}{8} (- ln tan (arctan (2 x)) + sec (arctan (2 x)) + \frac{1}{2} sec (arctan (2 x)) tan (arctan (2 x)) + \frac{1}{2} ln tan (arctan (2 x)) + sec (arctan (2 x))) : - \frac{1}{2} ln 2 x + 4 x + 1 + x 4 x + 1

= - \frac{1}{2} ln 2 x + 4 x + 1 + x 4 x + 1

Add a constant to the solution

= - \frac{1}{2} ln 2 x + 4 x + 1 + x 4 x + 1 + C

9 cos^{2} (t)

x^{3} - 1

\frac{\partial}{\partial x} (e^{y} + x + x^{2} + 5)

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

y = 3 t + 4 t^{3}

What is the integral of (4sqrt(x))/(sqrt(4x+1)) ?
The integral of (4sqrt(x))/(sqrt(4x+1)) is -1/2 ln|2sqrt(x)+sqrt(4x+1)|+sqrt(x)sqrt(4x+1)+C