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Popular >

integral of x/(1+x)

Solution

\int \frac{x}{1 + x} d x

Solution

1 + x - ln ∣ 1 + x ∣ + C

Solution steps

\int \frac{x}{1 + x} d x

Apply u-substitution

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

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

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = 1 + x

= 1 + x - ln ∣ 1 + x ∣

Add a constant to the solution

= 1 + x - ln ∣ 1 + x ∣ + C

Popular Examples

y^'=1+x-y-xy,y(0)=-1

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

limit as x approaches pi of-4sin(x/2)

x \to π lim (- 4 sin (\frac{x}{2}))

(1 - x^{2}) y^{^{'}} = 1

(\partial)/(\partial x)(25x^8y^4+12x^4y^5)

\frac{\partial}{\partial x} (25 x^{8} y^{4} + 12 x^{4} y^{5})

integral of 5e^x+cos(x/2)

\int 5 e^{x} + cos (\frac{x}{2}) d x

Frequently Asked Questions (FAQ)

What is the integral of x/(1+x) ?
The integral of x/(1+x) is 1+x-ln|1+x|+C