Advertisements
Advertisements
Question
Evaluate: ∫ x . log x dx
Advertisements
Solution
Let ∫ x . log x dx
= ∫ log x . x . dx ...[by LIATE rule]
Integrating by parts
I = `"log x" . int "x" "dx" - int ["d"/"dx" ("log x") int "x" "dx"] "dx"`
`= "log x" . ("x"^2/2) - int 1/"x" . ("x"^2/2) "dx"`
`= "x"^2/2 . "log x" - 1/2 int "x" "dx"`
`= "x"^2/2 . "log x" - 1/2 ("x"^2/2) + "c"`
`= "x"^2/2 . "log x" - "x"^2/4" + c`
APPEARS IN
RELATED QUESTIONS
Write the following compound statement symbolically.
Nagpur is in Maharashtra and Chennai is in Tamil Nadu.
Construct the truth table of the following statement pattern.
∼ p ∧ [(p ∨ ∼ q) ∧ q]
Express the following statement in symbolic form.
e is a vowel or 2 + 3 = 5
Express the following statement in symbolic form.
Mango is a fruit but potato is a vegetable.
Write the truth value of the following statement.
Earth is a planet and Moon is a star.
Write the following statement in symbolic form.
It is not true that “i” is a real number.
Write the following statement in symbolic form.
Milk is white if and only if the sky is not blue.
Find the truth value of the following statement.
Neither 27 is a prime number nor divisible by 4.
Write the negation of the following.
Kanchanganga is in India and Everest is in Nepal.
Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.
Stock prices are high or stocks are not rising iff stocks are rising.
Write the negation of the following statement.
∃ x ∈ A, such that x + 5 < 11.
If p → q is an implication, then the implication ∼ q → ∼ p is called its
Write the following statement in symbolic form:
Milk is white if and only if the sky is not blue.
Given 'p' and 'q' as true and 'r' as false, the truth values of p v (q ∧ ~r) and (p → r) ∧ q are respectively
Which of the following is NOT true for p → q.
Conditional of p → q is equivalent to p → ∼ q.
Let p, q and r be any three logical statements. Which of the following is true?
Express the following compound statement symbolically:
Delhi is in India but Dhaka is not in Sri Lanka
Construct the truth table for the statement pattern:
[(p → q) ∧ q] → p
