UNIT1 | unit 1 calculus

MATHS I

UNIT-1

1.1Evaluation of definite and improper integrals.

1.2 Reduction formulae.

1.3 Beta and Gamma functions and their properties.

1.4 Applications of definite integrals to evaluate surface areas and volume of revolutions.

1.1Evaluation of definite and improper integrals.

An improper integral is a definite integral—one with upper and lower limits—that goes to infinity in one direction or another. Either one of its limits are infinity, or the integrand (that function inside the interval, usually represented by f(x)) goes to infinity in the integral.

Improper integrals are integrals you can’t immediately solve because of the infinite limit(s) or vertical asymptote in the interval. The reason you can’t solve these integrals without first turning them into a proper integral (i.e. one without infinity) is that in order to integrate, you need to know the interval length. And if your interval length is infinity, there’s no way to determine that interval. The workaround is to turn the improper integral into a proper one and then integrate by turning the integral into a limit problem.

Example1: Solve the following improper Integral

Solution:

Given,

Example 2: Solve the following improper Integral

Solution: Given,

Example 3: Solve the following improper Integral

Solution: Given,

Example 4 : Solve the following improper Integral

Solution: Given,

1.2 Reduction formulae.

Reduction formula is regarded as a method of integration. Integration by reduction formula helps to solve the powers of elementary functions, polynomials of arbitrary degree, products of transcendental functions and the functions that cannot be integrated easily, thus, easing the process of integration and its problems.

Reduction Formula for Exponential Functions

∫xn emx dx = [(1/m) xnemx ]− [(n/m) ∫xn−1 emx ]dx

∫ dx = − []+ [() ∫dx, n≠1

Reduction Formula for Hyperbolic Trigonometric Functions

∫sinhnx dx = − sinhn−1 x cosh x − ∫sinhn−2x dx
∫coshnx dx = sinhx coshn−1 x − ∫coshn−2x dx
∫tanhnx dx = − tanhn−1x + ∫tanhn−2x dx, n≠1
∫cothnx dx = − cothn−1x + ∫cothn−2x dx, n≠1

Reduction formulae for Trigonometric Functions

Reduction Formula for Logarithmic Functions:

Example 1: evaluate the integral

Solution: we use

Substituting a=2x

Hence,

Example2: evaluate the integral

Solution: we use the formula

= .

1.3 Beta and Gamma functions and their properties.

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

{\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt}