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.
If ΔABC is right-angled at B, then m∠A + m∠C = 90°.
Write the following compound statement symbolically.
x is not irrational number but is a square of an integer.
Construct the truth table of the following statement pattern.
(q → p) ∨ (∼ p ↔ q)
If p : He swims
q : Water is warm
Give the verbal statement for the following symbolic statement.
q → p
Assuming the first statement p and second as q. Write the following statement in symbolic form.
The Sun has set and Moon has risen.
Assuming the first statement p and second as q. Write the following statement in symbolic form.
Kavita is brilliant and brave.
Assuming the first statement p and second as q. Write the following statement in symbolic form.
x3 + y3 = (x + y)3 if xy = 0.
Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.
p→(q ∨ r)
Rewrite the following statement without using conditional –
(Hint : p → q ≡ ∼ p ∨ q)
If price increases, then demand falls.
Write the negation of the following.
An angle is a right angle if and only if it is of measure 90°.
Rewrite the following statement without using the connective ‘If ... then’.
If a quadrilateral is rhombus then it is not a square.
Write the negation of the following statement.
7 is prime number and Tajmahal is in Agra.
The negation of the statement (p ˄ q) `→` (r ˅ ∼ p) is ______.
Which of the following is NOT true for p → q.
Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."
The statement ∼(p ↔ ∼q) is ______.
Write the contrapositive of the inverse of the statement:
‘If two numbers are not equal, then their squares are not equal’.
Using the statements
p: Seema is fat,
q: Seema is happy,
Write the following statements in symbolic form;
- Seema is thin and happy.
- If Seema is fat then she is unhappy.
Let p: I am brave.
q: I will climb the Mount Everest.
The symbolic form of a statement,
‘I am neither brave nor I will climb the mount Everest’ is
