The value of $\int \frac{e^{6 \log x} -e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} dx$ is equal to
- $0 + C$
- $\frac {x^3}{3} + C$
- $\frac {3}{x^3} +C$
- $\frac {1}{x} +C $
The Correct Option is B
Solution and Explanation
$=\int \frac{x^{6}-x^{5}}{x^{4}-x^{3}} d x \left[\because e^{y \log x}=x^{y}\right]$
$=\int \frac{x^{5}(x-1)}{x^{3}(x-1)} d x=\int x^{2} d x $
$=\frac{x^{3}}{3}+C $
Top Questions on Methods of Integration
- Evaluate the integral \(\int_{0}^{1} \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx\) Given that the integral can be expressed in the form \(a + b\sqrt{2} + c\sqrt{3}\), where \(a, b, c\) are rational numbers, find the value of \(2a + 3b - 4c\).
- JEE Main - 2024
- Mathematics
- Methods of Integration
- If $\phi(x)=\frac{1}{\sqrt{ x }} \int_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) dt , x>$, then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :
- JEE Main - 2023
- Mathematics
- Methods of Integration
- Let f:[1,∞) \(\rightarrow\) R be a differentiable function such that f(1) = \(\frac{1}{3}\) and 3\(\int_{1}^{x}\)f(t)dt=xf(x)-\(\frac{x^3}{3}\),x∈[1,∞). Let e denote the base of the natural logarithm. Then the value of f(e) is
- JEE Advanced - 2023
- Mathematics
- Methods of Integration
Let f : (0,1) → R be the function defined as f(x) = √n if x ∈ [\(\frac{1}{n+1},\frac{1}{n}\)] where n ∈ N. Let g : (0,1) → R be a function such that \(\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt x\) for all x ∈ (0,1).
Then \(\lim_{x\rightarrow0}f(x)g(x)\)- JEE Advanced - 2023
- Mathematics
- Methods of Integration
- Let n ≥ 2 be a natural number and f:[0,1)]\(\rightarrow\) R be the function defined by
If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is- JEE Advanced - 2023
- Mathematics
- Methods of Integration
Questions Asked in KCET exam
- Ten identical cells each emf 2V and internal resistance 1Ω are connected in series with two cells wrongly connected. A resistor of 10Ω is connected to the combination. What is the current through the resistor?
- KCET - 2023
- Cells in Series and in Parallel
- The speed of sound in an ideal gas at a given temperature T is v. The rms speed of gas molecules at that temperature is vrms. The ratio of the velocities v and vrms for helium and oxygen gases are X and X' respectively. Then \(\frac{X}{X'}\) is equal to
- KCET - 2023
- Thermodynamics
- If \(p(\frac{1}{q}+\frac{1}{r}),q(\frac{1}{r}+\frac{1}{p}),r(\frac{1}{p}+\frac{1}{q})\) are in A.P., then p, q, r
- KCET - 2023
- Arithmetic Progression
- A stretched wire of a material whose Young's modulus Y = 2 × 1011 Nm-2 has Poisson's ratio of 0.25. Its lateral strain εl = 10-3. The elastic energy density of the wire is
- KCET - 2023
- mechanical properties of solids
- A metallic rod of length 1 m held along east-west direction is allowed to fall down freely. Given horizontal component of earth’s magnetic field BH = 3 × 10-5 T. The emf induced in the rod at an instant t = 2s after it is released is ( Take g = 10 ms-2 )
- KCET - 2023
- Faradays laws of induction
Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C