What is expand (1+2sin(x))^5 ?

The solution to expand (1+2sin(x))^5 is 1+10sin(x)+40sin^2(x)+80sin^3(x)+80sin^4(x)+32sin^5(x)

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expand (1+2sin(x))^5

expand

(1 + 2 sin (x))^{5}

1 + 10 sin (x) + 40 sin^{2} (x) + 80 sin^{3} (x) + 80 sin^{4} (x) + 32 sin^{5} (x)

Solution steps

(1 + 2 sin (x))^{5}

Apply binomial theorem:

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

a = 1, b = 2 sin (x)

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (2 sin (x))^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (2 sin (x))^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (2 sin (x))^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (2 sin (x))^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (2 sin (x))^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (2 sin (x))^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (2 sin (x))^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (2 sin (x))^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (2 sin (x))^{1} : 10 sin (x)

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (2 sin (x))^{2} : 40 sin^{2} (x)

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (2 sin (x))^{3} : 80 sin^{3} (x)

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (2 sin (x))^{4} : 80 sin^{4} (x)

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (2 sin (x))^{5} : 32 sin^{5} (x)

= 1 + 10 sin (x) + 40 sin^{2} (x) + 80 sin^{3} (x) + 80 sin^{4} (x) + 32 sin^{5} (x)

(1 + 2 sin (x))^{2}

\frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

(2, 1 + i, i) (2 - i, 2, 1 + 2 i)

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

(x^{- 1})^{2}

What is expand (1+2sin(x))^5 ?
The solution to expand (1+2sin(x))^5 is 1+10sin(x)+40sin^2(x)+80sin^3(x)+80sin^4(x)+32sin^5(x)