Evaluate the definite integral: \(∫^\frac{π}{2}_0 cos^2 x dx\)
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
\(I=[F(\frac{π}{2})-F(0)]\)
\(=\frac{1}{2}[(\frac{π}{2})-\frac{sinπ}{2})-(0+\frac{sin0}{2})]\)
\(=\frac{1}{2}[\frac{π}{2}+0-0-0]\)
\(=\frac{π}{4}\)
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?
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.