Question:
$\int \frac{dx}{(x + a)^{9/7} (x - b)^{5/7}}$ = ______.
$\int \frac{dx}{(x + a)^{9/7} (x - b)^{5/7}}$ = ______.
Show Hint
If the sum of the powers of the two linear factors in the denominator is exactly 2, divide and multiply one factor by the other to create a $(\text{linear}/\text{linear})^n$ term and an $(x+k)^2$ term. The derivative of a linear fraction is always a constant over a square!
Updated On: Jun 19, 2026
- $(7/(a + b)) [(x - b)/(x + a)]^{9/7} + c$, where c is the constant of integration
- $(7/(a + b)) [(x - b)/(x + a)]^{5/7} + c$, where c is the constant of integration
- $(7/(2(a + b))) [(x - b)/(x + a)]^{2/7} + c$, where c is the constant of integration
- $(7/(a + b)) [(x - b)/(x + a)]^{1/7} + c$, where c is the constant of integration
Note: Option (c) frequently appears in OCR text as $9/7$ due to poor print quality, but mathematically evaluates to $2/7$.
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We must integrate a fractional algebraic expression where the sum of the fractional powers in the denominator is an integer: $9/7 + 5/7 = 14/7 = 2$.
Step 2: Key Formula or Approach:
When integrating $\frac{dx}{(x-\alpha)^m (x-\beta)^n}$ where $m+n=2$, the standard algebraic trick is to force the terms into a single rational function raised to a power. We do this by factoring out one of the terms to create a perfect square denominator.
Step 3: Detailed Explanation:
Rewrite the integral by extracting $(x+a)^2$ from the denominator:
$$\int \frac{dx}{(x+a)^{9/7} (x-b)^{5/7}} = \int \frac{dx}{(x+a)^2 \cdot \frac{(x-b)^{5/7}}{(x+a)^{5/7}}}$$
$$= \int \frac{dx}{(x+a)^2 \left( \frac{x-b}{x+a} \right)^{5/7}}$$
Now, perform substitution. Let $t = \frac{x-b}{x+a}$.
Find $dt$ using the quotient rule:
$$dt = \frac{(x+a)\frac{d}{dx}(x-b) - (x-b)\frac{d}{dx}(x+a)}{(x+a)^2} dx$$
$$dt = \frac{(x+a)(1) - (x-b)(1)}{(x+a)^2} dx = \frac{a+b}{(x+a)^2} dx$$
So, $\frac{dx}{(x+a)^2} = \frac{dt}{a+b}$.
Substitute these components back into the modified integral:
$$= \int \frac{1}{t^{5/7}} \cdot \frac{dt}{a+b} = \frac{1}{a+b} \int t^{-5/7} dt$$
Integrate using the power rule:
$$= \frac{1}{a+b} \cdot \frac{t^{-5/7 + 1}}{-5/7 + 1} + c = \frac{1}{a+b} \cdot \frac{t^{2/7}}{2/7} + c$$
$$= \frac{7}{2(a+b)} t^{2/7} + c$$
Back-substitute $t$:
$$= \frac{7}{2(a+b)} \left( \frac{x-b}{x+a} \right)^{2/7} + c$$
Step 4: Final Answer:
The result matches option (c) (accounting for the OCR misread of the final power).
We must integrate a fractional algebraic expression where the sum of the fractional powers in the denominator is an integer: $9/7 + 5/7 = 14/7 = 2$.
Step 2: Key Formula or Approach:
When integrating $\frac{dx}{(x-\alpha)^m (x-\beta)^n}$ where $m+n=2$, the standard algebraic trick is to force the terms into a single rational function raised to a power. We do this by factoring out one of the terms to create a perfect square denominator.
Step 3: Detailed Explanation:
Rewrite the integral by extracting $(x+a)^2$ from the denominator:
$$\int \frac{dx}{(x+a)^{9/7} (x-b)^{5/7}} = \int \frac{dx}{(x+a)^2 \cdot \frac{(x-b)^{5/7}}{(x+a)^{5/7}}}$$
$$= \int \frac{dx}{(x+a)^2 \left( \frac{x-b}{x+a} \right)^{5/7}}$$
Now, perform substitution. Let $t = \frac{x-b}{x+a}$.
Find $dt$ using the quotient rule:
$$dt = \frac{(x+a)\frac{d}{dx}(x-b) - (x-b)\frac{d}{dx}(x+a)}{(x+a)^2} dx$$
$$dt = \frac{(x+a)(1) - (x-b)(1)}{(x+a)^2} dx = \frac{a+b}{(x+a)^2} dx$$
So, $\frac{dx}{(x+a)^2} = \frac{dt}{a+b}$.
Substitute these components back into the modified integral:
$$= \int \frac{1}{t^{5/7}} \cdot \frac{dt}{a+b} = \frac{1}{a+b} \int t^{-5/7} dt$$
Integrate using the power rule:
$$= \frac{1}{a+b} \cdot \frac{t^{-5/7 + 1}}{-5/7 + 1} + c = \frac{1}{a+b} \cdot \frac{t^{2/7}}{2/7} + c$$
$$= \frac{7}{2(a+b)} t^{2/7} + c$$
Back-substitute $t$:
$$= \frac{7}{2(a+b)} \left( \frac{x-b}{x+a} \right)^{2/7} + c$$
Step 4: Final Answer:
The result matches option (c) (accounting for the OCR misread of the final power).
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Integration Questions
- Evaluate the integral: \[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]
- Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]
- Evaluate the integral: \[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}\,dx \]
- Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \cos x} \]
- Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions