What is the integral of x/(sqrt(25+x^2)) ?

The integral of x/(sqrt(25+x^2)) is sqrt(25+x^2)+C

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integral of x/(sqrt(25+x^2))

\int \frac{x}{25 + x ^{2}} d x

25 + x^{2} + C

Solution steps

\int \frac{x}{25 + x ^{2}} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \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 \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 25 + x^{2}

= \frac{1}{2} \cdot 2 (25 + x^{2})^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (25 + x^{2})^{\frac{1}{2}} : 25 + x^{2}

= 25 + x^{2}

Add a constant to the solution

= 25 + x^{2} + C

f (x) = 2 x^{3} + 3 x^{2} - 12 x + 5

n \to \infty lim (n - 1)

\frac{d y}{d x} = \frac{x ^{2}}{y ( 1 + x ^{3} )}

y = \frac{4 x + 1}{x}

What is the integral of x/(sqrt(25+x^2)) ?
The integral of x/(sqrt(25+x^2)) is sqrt(25+x^2)+C