Integrate the function: \(\frac{1}{\cos^2x(1-\tan x)^2}\)
Solution and Explanation
\(\frac{1}{\cos^2x(1-\tan x)^2}=\frac{\sec^2x}{(1-\tan x)^2}\)
Let (1-tanx) = t
∴ -sec2 xdx=dt
\(\Rightarrow \int\frac{\sec^2x}{(1-\tan x)^2}dx=\int\frac{-dt}{t^2}\)
= \(-\int t^2dt\)
= \(+\frac{1}{t}+C\)
=\(\frac{1}{(1-\tan x)}+C\)
Top Questions on integral
- Evaluate the integral: $$ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx $$
- Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
- Evaluate the integral \( \int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx \)
- Evaluate the integral: $$ \int_0^1 \frac{\ln (1+x)}{1+x^2} \, dx $$
- Evaluate $ \int_{0}^{1} (3x^2 + 2x) \, dx $.
Questions Asked in CBSE CLASS XII exam
- A bill of exchange is drawn on 1st July, 2021 payable one month after the date. What will be the date of its maturity?
- CBSE CLASS XII - 2025
- Banking Operations
- Indian labour market indicators have improved in the last six years, as per the Periodic Labour Force Survey Data. The unemployment rate has fallen to 3.2% in 2022–23. Rising youth and female participation in the workforce present an opportunity to tap the demographic and gender dividend. The factory employment data exhibits the bounce back of the organised manufacturing sector in FY 2022–23 with rise in employment and the upscaling of factories.
The net payroll additions under EPFO have more than doubled in the past five years, signalling healthy growth in formal employment.
To generate and sustain quality employment, agro-processing and care economy are two promising candidates.
On the basis of the given text and common understanding, answer the following questions:- CBSE CLASS XII - 2025
- Economic Reforms
Government provides certain goods and services which cannot be provided by the market mechanism. Examples of such goods are national defence, roads, government administration etc. which are referred to as public goods.
There are two major differences between public and private goods. One, the benefits of public goods are available to all and are not only restricted to one particular consumer. For example, if a person wears a shirt, it will not be available to others. It is said that this person’s consumption stands in rival relationship to the consumption of others. However, if we consider a public park or measures to reduce air pollution, the benefits will be available to all. One person’s consumption of a good does not reduce the amount available for consumption for others and so several people can enjoy the benefits, that is, the consumption of many people is not ’rivalrous’.
Two, in case of private goods, anyone who does not pay for the goods can be excluded from enjoying its benefits. If you do not buy a ticket, you will not be allowed to watch a movie at a local cinema hall. However, in case of public goods, there is no feasible way of excluding anyone from enjoying the benefits of the good. That is why public goods are called non-excludable. Even if some users do not pay, it is difficult and sometimes impossible to collect fees for the public good. These non-paying users are known as ’free-riders’. Consumers will not voluntarily pay for what they can get for free and for which there is no exclusive title to the property being enjoyed. The link between the producer and consumer which occurs through the payment process is broken and the government must step in to provide for such goods.- CBSE CLASS XII - 2025
- Public goods and market failure
- Growing carbon footprint of industries have put power and steel sector in the spotlight as the major contributor to the climate crisis. The challenge of climate change can be tackled only by making our industries and businesses follow practices and processes that reduce their carbon footprint. It can be possible only with green financing.
Green financing aims to increase the level of financial flows (from banking, micro-credit, insurance and investment) from the public, private and not-for-profit sectors to sustainable development priorities.
Global green finance has also started chasing Indian companies. Global development finance institutions and funds are ready to offer long-term support (both equity and debt) at cheap rates to projects like solar energy and hydropower.
Green finance can positively affect environment quality, economic mdevelopment and financial issues that promote the green economy, such as reducing greenhouse gas emissions, improving energy efficiency or enhancing the organic economy.
On the basis of the given text and common understanding, answer the following questions :- CBSE CLASS XII - 2025
- Indian Economy
- "Globalisation has been a process for India with only positive results."
Defend or refute the given statement with valid explanation.- CBSE CLASS XII - 2025
- Indian Economy
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