What is the integral from 0 to 1 of 1/(sqrt(x+x^2)) ?

The integral from 0 to 1 of 1/(sqrt(x+x^2)) is 2ln(1+sqrt(2))

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integral from 0 to 1 of 1/(sqrt(x+x^2))

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

2 ln (1 + 2)

Decimal

1.76274 \dots

Solution steps

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

Factor

\frac{1}{x + x ^{2}}

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (x + 1) = x (x + 1)

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

Apply u-substitution

= \int_{0}^{1} \frac{2}{u ^{2} + 1} d u

Take the constant out:

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

= 2 \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= 2 [ln u^{2} + 1 + u]_{0}^{1}

Compute the boundaries:

ln (1 + 2)

= 2 ln (1 + 2)

What is the integral from 0 to 1 of 1/(sqrt(x+x^2)) ?
The integral from 0 to 1 of 1/(sqrt(x+x^2)) is 2ln(1+sqrt(2))