Evaluate the definite integral: \(∫^\frac{π}{2}_0 cos^2 x dx\)
Evaluate the definite integral: \(∫^\frac{π}{2}_0 cos^2 x dx\)
Solution and Explanation
Let I=\(∫^\frac{π}{2}_0 cos^2 x dx\)
\(∫cos^2 x dx=∫(\frac{1+cos2x}{2})dx=\frac{x}{2}+\frac{sin2x}{4}=\frac{1}{2}(x+\frac{sin2x}{2})=F(x)\)
By second fundamental theorem of calculus,we obtain
\(I=[F(\frac{π}{2})-F(0)]\)
\(=\frac{1}{2}[(\frac{π}{2})-\frac{sinπ}{2})-(0+\frac{sin0}{2})]\)
\(=\frac{1}{2}[\frac{π}{2}+0-0-0]\)
\(=\frac{π}{4}\)
Top Questions on integral
- Evaluate the integral: $$ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx $$
- 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 $.
- Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]
Questions Asked in CBSE CLASS XII exam
1. Haemophilia and red-green colour-blindness is usually observed in men. Why?
2. Perform a cross (or crosses) where haemophilic daughter(s) and haemophilic son(s) are
produced in the same ratio.OR
1. Where do transcription and translation occur in bacteria and eukaryotes respectively?
2. Draw a labelled schematic sketch of replication fork of DNA.
3. A DNA segment has a total of 1000 nucleotides, out of which 240 of them are Adenine-containing nucleotides. How many pyrimidine bases does this segment possess?- CBSE CLASS XII - 2025
- Principles of Inheritance and Variation
- Name the restriction endonuclease that recognizes this sequence.
- CBSE CLASS XII - 2025
- Biotechnology
- Assertion (A): While working on Staphylococci, Alexander Fleming observed that Penicillium notatum inhibits the growth of bacteria.
Reason (R): The inhibiting chemical was commercially extracted and its full potential was established by Alexander Fleming.- CBSE CLASS XII - 2025
- human health and disease
- Differentiate between exonuclease and endonuclease.
- CBSE CLASS XII - 2025
- Biotechnology
- A Dihybrid cross is done between two parent pea plants (pure line) who differ in two pairs of contrasting traits: Seed colour and seed shape. In the F$_2$ generation the number of phenotypes and genotypes will be:
- CBSE CLASS XII - 2025
- Principles of Inheritance and Variation
Concepts Used:
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus is the theorem which states that differentiation and integration are opposite processes (or operations) of one another.
Calculus's fundamental theorem connects the notions of differentiating and integrating functions. The first portion of the theorem - the first fundamental theorem of calculus – asserts that by integrating f with a variable bound of integration, one of the antiderivatives (also known as an indefinite integral) of a function f, say F, can be derived. This implies the occurrence of antiderivatives for continuous functions.