What is expand (x+ln(x))^4 ?

The solution to expand (x+ln(x))^4 is x^4+4x^3ln(x)+6x^2ln^2(x)+4xln^3(x)+ln^4(x)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

expand (x+ln(x))^4

expand

(x + ln (x))^{4}

x^{4} + 4 x^{3} ln (x) + 6 x^{2} ln^{2} (x) + 4 x ln^{3} (x) + ln^{4} (x)

Solution steps

(x + ln (x))^{4}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = ln (x)

= i = 0 \sum 4 (i 4) x^{(4 - i)} ln^{i} (x)

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} ln^{0} (x) + \frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} ln^{1} (x) + \frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} ln^{2} (x) + \frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} ln^{3} (x) + \frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} ln^{4} (x)

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} ln^{0} (x) : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} ln^{1} (x) : 4 x^{3} ln (x)

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} ln^{2} (x) : 6 x^{2} ln^{2} (x)

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} ln^{3} (x) : 4 x ln^{3} (x)

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} ln^{4} (x) : ln^{4} (x)

= x^{4} + 4 x^{3} ln (x) + 6 x^{2} ln^{2} (x) + 4 x ln^{3} (x) + ln^{4} (x)

- \frac{3 ( x + 3 ) ( x - 1 )}{x ^{2}}

3 m^{2 (m - n) (2 m + n)}

\frac{x ^{2} + 1}{x ( x - 1 ) ( x + 1 ) ( x - 2 )}

\frac{100 x ^{3} - 600 x ^{2} + 1200 x - 800}{x}

\frac{4 x}{( x - 1 ) ^{2}}

What is expand (x+ln(x))^4 ?
The solution to expand (x+ln(x))^4 is x^4+4x^3ln(x)+6x^2ln^2(x)+4xln^3(x)+ln^4(x)