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### Integral Calculus Problem Set III - Examples and solved problems related to Reduction Formulas, Improper Integrals, Other interesting definite and indefinite integrals

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## Integral Calculus- Problem Set III ## Calculus - Integral Calculus Problem Set III - Outline of Contents:

Examples and solved problems - Reduction formulas, improper integrals, reducing the integrand to partial fractions, more of definite integrals.

Here are the kind of cases we will cover in this tutorial.

1
∫ log x dx    -- Solve this Integral. We eventually simplify this to the limiting value of [−h log h − (1 − h)] as h->0

Showing that the area bound by the curves  eax sin(bx) and  eax cos bx, and the positive x-axis is b/(a2 + b2) and a/(a2 + b2) respectively.
We solve these using integration by parts, and then using limiting values (as x->infinity)

Establish a reduction formula for   ∫ xn eax dx and apply it to evaluate  ∫  x3 eax dx.
We solve this using integration by parts, to arrive at a reduction formula, and then put n = 1, 2, 3 respectively

Obtain a reduction formula for ∫ xm sin nx dx.
Apply integration of parts. You get a reduction formula including a term of ∫ xm-1 cos nx dx, on which you again apply integration of parts to get a formula including the term  ∫ xm-2 sin nx dx.

Obtain a reduction formula for  ∫ xn e−x dx  and show that the value equals n! when we compute the value of the definite integral  between x = 0 and infinity.
We solve this using integration by parts. One term involves a multiple of ∫ xn-1 e−x dx  and the other one can be eliminated altogeter because it tends to 0

Obtain a reduction formula for  ∫  xm (log x)n dx and use it to evaluate
1
x4 (log x)3 dx.
0

We also pick up other interesting problems involving polynomials in the denominator. These are solved using partial fractions.