My attempt: \begin{align} \int{\frac{x-\sqrt{x^2-5x+6}}{x+\sqrt{x^2+5x+6}}}dx &\\ &=\int{\frac{x+\sqrt{x^2+5x+6}}{x+\sqrt{x^2+5x+6}}}dx+\int{\frac{-\sqrt{x^2+5x+6}-\sqrt{x^2+5x+6}}{x+\sqrt{x^2-5x+6}}}dx \\ &=x-\int{\frac{\sqrt{x^2+5x+6}}{x+\sqrt{x^2+5x+6}}}dx-\int{\frac{\sqrt{x^2-5x+6}}{x+\sqrt{x^2+5x+6}}}dx \\ &=x-I_1-I_2 \end{align} I know how to solve the integral $I_1$ using Euler substitution, but I don't know how to solve the integral $I_2$.
2026-04-06 16:18:59.1775492339
Solve the integral $ \int{\frac{x-\sqrt{x^2-5x+6}}{x+\sqrt{x^2+5x+6}}}dx $
158 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in CALCULUS
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- Proving the differentiability of the following function of two variables
- If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Number of roots of the e
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- How to prove $\frac 10 \notin \mathbb R $
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
Related Questions in INDEFINITE-INTEGRALS
- Closed form of integration
- How to find $\int \sqrt{x^8 + 2 + x^{-8}} \,\mathrm{d}x$?
- Find the integral $\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx.$
- Integrate $\int \frac {x^4}{\sqrt {x^2-9}} \,dx$
- Integral of $\frac{1}{2x}$.
- Contradictory results of the integral of an odd function
- Integrate $\int \frac{x+2}{(x^2+3x+3) \sqrt{x+1}} dx$
- Evaluation of Integral $\int \frac{x^2+1}{\sqrt{x^3+3}}dx$
- Integral of a Polynomial in Square Root
- Using a substitution of a square of a trigonometric function.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Hint:
$\int\dfrac{x-\sqrt{x^2-5x+6}}{x+\sqrt{x^2+5x+6}}~dx$
$=\int\dfrac{x-\sqrt{(x-3)(x-2)}}{\sqrt{(x+3)(x+2)}+x}~dx$
$=\int\dfrac{\left(x-\sqrt{(x-3)(x-2)}\right)\left(\sqrt{(x+3)(x+2)}-x\right)}{\left(\sqrt{(x+3)(x+2)}+x\right)\left(\sqrt{(x+3)(x+2)}-x\right)}~dx$
$=\int\dfrac{x\sqrt{(x+3)(x+2)}}{5x+6}~dx+\int\dfrac{x\sqrt{(x-3)(x-2)}}{5x+6}~dx-\int\dfrac{x^2}{5x+6}~dx-\int\dfrac{\sqrt{(x+3)(x+2)}\sqrt{(x-3)(x-2)}}{5x+6}~dx$