What is the integral from 0 to 1 of (1+sqrt(x))^6 ?

The integral from 0 to 1 of (1+sqrt(x))^6 is 769/28

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integral from 0 to 1 of (1+sqrt(x))^6

\int_{0}^{1} (1 + x)^{6} d x

\frac{769}{28}

Decimal

27.46428 \dots

Solution steps

\int_{0}^{1} (1 + x)^{6} d x

Expand

(1 + x)^{6} : 1 + 6 x + 15 x + 20 x^{\frac{3}{2}} + 15 x^{2} + 6 x^{\frac{5}{2}} + x^{3}

= \int_{0}^{1} 1 + 6 x + 15 x + 20 x^{\frac{3}{2}} + 15 x^{2} + 6 x^{\frac{5}{2}} + x^{3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} 1 d x + \int_{0}^{1} 6 x d x + \int_{0}^{1} 15 x d x + \int_{0}^{1} 20 x^{\frac{3}{2}} d x + \int_{0}^{1} 15 x^{2} d x + \int_{0}^{1} 6 x^{\frac{5}{2}} d x + \int_{0}^{1} x^{3} d x

\int_{0}^{1} 1 d x = 1

\int_{0}^{1} 6 x d x = 4

\int_{0}^{1} 15 x d x = \frac{15}{2}

\int_{0}^{1} 20 x^{\frac{3}{2}} d x = 8

\int_{0}^{1} 15 x^{2} d x = 5

\int_{0}^{1} 6 x^{\frac{5}{2}} d x = \frac{12}{7}

\int_{0}^{1} x^{3} d x = \frac{1}{4}

= 1 + 4 + \frac{15}{2} + 8 + 5 + \frac{12}{7} + \frac{1}{4}

Simplify

1 + 4 + \frac{15}{2} + 8 + 5 + \frac{12}{7} + \frac{1}{4} : \frac{769}{28}

= \frac{769}{28}

\int_{0}^{2} (2 x + 3)^{3} d x

\int_{0}^{1} 12 x^{2} y^{3} d y

\int_{0}^{4} 2 x^{3} + 1 d x

\int_{0}^{6} \frac{1}{1 + x ^{2}} d x

\int_{0}^{\frac{π}{6}} (sec (x) + tan (x))^{2} d x

What is the integral from 0 to 1 of (1+sqrt(x))^6 ?
The integral from 0 to 1 of (1+sqrt(x))^6 is 769/28