∫ (x2 - 1)dx/(x3(2x4 - 2x2 1)1/2) = ?
∫ (x2 - 1)dx/(x3(2x4 - 2x2 1)1/2) = ?
Solution and Explanation
The integral of (x^2 - 1)dx / (x^3(2x^4 - 2x^2 + 1)^(1/2)) is equal to -(2x^4 - 2x^2 + 1)^(-1/2) + C, where C is the constant of integration.
Top Questions on Methods of Integration
- For any y∈R, let cot\(^{-1}\)(y) ∈ (0,π) and tan\(^{-1}\)(y) ∈ (\(-\frac{\pi}{2},\frac{\pi}{2}\)). Then the sum of all the solutions of the equation\(tan^{-1}(\frac{6y}{9-y^2})+cot^{-1}(\frac{9-y^2}{6y})=\frac{2\pi}{3}\, for\,0<|y|<3,\) is equal to
- JEE Advanced - 2023
- 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 : (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
\(\int \frac{1}{(x + 2)(1 + x)^2} \, dx\)
- MHT CET - 2023
- Mathematics
- Methods of Integration
- Let S be the set of all twice differentiable functions f from R to R such that \(\frac{d^2f}{dx^2}(x)>0\) for all x∈(-1,1). For f∈S, let Xf be the number of points x∈(-1,1) for which f(x)=x.Then which of the following statements is(are) true?
- JEE Advanced - 2023
- Mathematics
- Methods of Integration
Questions Asked in MHT CET exam
- Two monkeys off mass 10 kg and 8 kg are moving along a vertical light rope the former climbing up with an acceleration of 2 m/second square while the latter coming down with a uniform velocity of 2 m/sec square find the tension in the rope at the fixed support
- MHT CET - 2024
- tension
- Total genetic content of an organism is called
- MHT CET - 2024
- Non-Mendelian Genetics
- How many ATP molecules are needed as an initial investment in the glycolytic cycle (normal glycolysis)?
- MHT CET - 2024
- Glycolysis
- Which disease is primarily spread by female Anopheles mosquitoes?
- MHT CET - 2024
- HIV and AIDS
- Find k if \(\int_{0}^{\frac{1}{2}}[\frac{x^2dx}{(1-x^2)^{\frac{3}{2}}}]=\frac{k}{6}\).
- MHT CET - 2023
- Definite Integral
MHT CET Notification
MHT CET 2024 CAP Round 2 Seat Allotment Out, Direct Link here.Aug 26, 2024Concepts 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
Integration Using Trigonometric Identities
SUBSCRIBE TO OUR NEWS LETTER
