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

The integral from 1 to N of x/(sqrt(1+x^2)) is sqrt(N^2+1)-sqrt(2)

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

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

N^{2} + 1 - 2

Solution steps

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

Apply u-substitution

: \int_{2}^{1 + N^{2}} \frac{1}{2 u} d u

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

Take the constant out:

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

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

Apply the Power Rule

: [2 u^{\frac{1}{2}}]_{2}^{1 + N^{2}}

= \frac{1}{2} [2 u^{\frac{1}{2}}]_{2}^{1 + N^{2}}

Apply radical rule:

= \frac{1}{2} [2 u]_{2}^{1 + N^{2}}

Compute the boundaries:

2 1 + N^{2} - 22

= \frac{1}{2} (2 1 + N^{2} - 22)

Simplify

= N^{2} + 1 - 2

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