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

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

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

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

x^{2} + 25 + C

Solution steps

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

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

Simplify

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

= x^{2} + 25

Add a constant to the solution

= x^{2} + 25 + C

y = cos (x), y = 1 - cos (x), 0 \leq x \leq π

\int \frac{24}{1 + 25 r ^{2}} d r

2 e^{3 t}

8 (\frac{1}{s ^{5}})

y = x^{3} + 2 x, y = 3 x

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