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Popular >

integral of x/(x+5)

Solution

\int \frac{x}{x + 5} d x

Solution

x + 5 - 5 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x}{x + 5} d x

Apply u-substitution

: \int \frac{u - 5}{u} d u

= \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \int 1 - \frac{5}{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{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u - 5 ln ∣ u ∣

Substitute back

u = x + 5

= x + 5 - 5 ln ∣ x + 5 ∣

Add a constant to the solution

= x + 5 - 5 ln ∣ x + 5 ∣ + C

Popular Examples

integral of (x-2)(3x-4)

\int (x - 2) (3 x - 4) d x

integral of 3ysin(xy)

\int 3 y sin (x y) d x

(dy)/(dx)=sqrt(7x^5+6)

\frac{d y}{d x} = 7 x^{5} + 6

derivative of f(x)=7e^{-x}-4e^{-7x}derivative of

f (x) = 7 e^{- x} - 4 e^{- 7 x}

(\partial)/(\partial x)(3sec^2(3x-2y+1))

\frac{\partial}{\partial x} (3 sec^{2} (3 x - 2 y + 1))

Frequently Asked Questions (FAQ)

What is the integral of x/(x+5) ?
The integral of x/(x+5) is x+5-5ln|x+5|+C