What is the integral of x/(sqrt(100x^2-1)) ?

The integral of x/(sqrt(100x^2-1)) is 1/100 sqrt(100x^2-1)+C

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integral of x/(sqrt(100x^2-1))

\int \frac{x}{100 x ^{2} - 1} d x

\frac{1}{100} 100 x^{2} - 1 + C

Solution steps

\int \frac{x}{100 x ^{2} - 1} d x

Apply u-substitution

: \int \frac{1}{200 u} d u

= \int \frac{1}{200 u} d u

Take the constant out:

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

= \frac{1}{200} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= \frac{1}{200} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 100 x^{2} - 1

= \frac{1}{200} \cdot 2 (100 x^{2} - 1)^{\frac{1}{2}}

Simplify

\frac{1}{200} \cdot 2 (100 x^{2} - 1)^{\frac{1}{2}} : \frac{1}{100} 100 x^{2} - 1

= \frac{1}{100} 100 x^{2} - 1

Add a constant to the solution

= \frac{1}{100} 100 x^{2} - 1 + C

\frac{d y}{d x}, x = y^{18}

g (x) = 1 - x

y^{^{''}} + y = 0, y (0) = 1, y^{^{'}} (0) = 0

x y^{^{''}} + y^{^{'}} = 20 x

\frac{\partial}{\partial y} (e^{2 x y^{2}} + 3 x)

What is the integral of x/(sqrt(100x^2-1)) ?
The integral of x/(sqrt(100x^2-1)) is 1/100 sqrt(100x^2-1)+C