Evaluate: ∫ sec 2 ( log x ) x d x

Given, ∫ sec 2 ( log x ) x d x Let, log x = t ⇒ 1 x d x = d t ⇒ ∫ sec 2 ( log x ) x d x = ∫ sec 2 t d t = tan t + c ∴ ∫ sec 2 ( log x ) x d x = tan log x + c

Evaluate: ∫ 2 x + 1 x 2 + x + 1 d x

Given, ∫ 2 x + 1 x 2 + x + 1 d x Let, x 2 + x + 1 = t ⇒ 2 x + 1 d x = d t ⇒ ∫ 2 x + 1 x 2 + x + 1 d x = ∫ t d t = 2 3 t 3 2 + c ∴ ∫ 2 x + 1 x 2 + x + 1 d x = 2 3 x 2 + x + 1 3 2 + c

Evaluate: ∫ d x 9 x 2 + 16

Given ∫ d x 9 x 2 + 16 = ∫ d x ( 3 x ) 2 + 4 2 We know that, ∫ d x ( a x ) 2 + b 2 = 1 a log a x + ( a x ) 2 + b 2 + c ∴ ∫ d x 9 x 2 + 16 = 1 3 log 3 x + 9 x 2 + 16 + c

Evaluate: ∫ d x 4 - 9 x 2

Given, ∫ d x 4 - 9 x 2 = ∫ d x 2 2 - ( 3 x ) 2 We know that, ∫ d x b 2 - ( a x ) 2 = 1 a sin - 1 a x b + c ∴ ∫ d x 4 - 9 x 2 = 1 3 sin - 1 3 x 2 + c

Evaluate: ∫ d x 4 x 2 - 25

Given ∫ d x 4 x 2 - 25 = ∫ d x ( 2 x ) 2 - 5 2 We know that, ∫ d x ( a x ) 2 - b 2 = 1 a log a x + ( a x ) 2 - b 2 + c ∴ ∫ d x 4 x 2 - 25 = 1 2 log 2 x + 4 x 2 - 25 + c

Evaluate: ∫ 4 - x 2 d x

Given ∫ 4 - x 2 d x = ∫ 2 2 - x 2 d x We know that, ∫ a 2 - x 2 d x = x 2 a 2 - x 2 + a 2 2 sin - 1 x a + c ∴ ∫ 4 - x 2 d x = x 2 4 - x 2 + 2 sin - 1 x 2 + c

Evaluate: ∫ 9 + x 2 d x

Given ∫ 9 + x 2 d x = ∫ 3 2 + x 2 d x We know that, ∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 log x + a 2 + x 2 + c ∴ ∫ 9 + x 2 d x = x 2 9 + x 2 + 9 2 log x + 9 + x 2 + c

Evaluate: ∫ x 2 - 16 d x

Given ∫ x 2 - 16 d x = ∫ x 2 - 4 2 d x We know that, ∫ x 2 - a 2 d x = x 2 x 2 - a 2 - a 2 2 log x + x 2 - a 2 + c ∴ ∫ x 2 - 16 d x = x 2 x 2 - 16 - 8 log x + x 2 - 16 + c

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Integral Calculus>Chapter 15 - Integration Using Partial Fractions>EXERCISE

R S Aggarwal Solutions for Exercise 2: EXERCISE

Author:R S Aggarwal

R S Aggarwal Mathematics Solutions for Exercise - R S Aggarwal Solutions for Exercise 2: EXERCISE

Attempt the free practice questions from Exercise 2: EXERCISE with hints and solutions to strengthen your understanding. SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12 solutions are prepared by Experienced Embibe Experts.