_1^4 x^3 x dx is equal to: Option 1: 644-256/16 Option 2: 1282+255/16 Option 3: 1282-255/16 Option 4: 642-255/4

Answer: 1282-255/16. Solution: Integrating by parts with x as the first function gives x^4/4 x-x^4/16. At x=4: 644-16; at x=1: -1/16. The difference is 1282-255/16, using 644=1282.

Applied Mathematics

Calculus

Medium

applied

$\int_1^4 x^3\log x\, dx$ is equal to:

$64\log4-\frac{256}{16}$

$128\log2+\frac{255}{16}$

$128\log2-\frac{255}{16}$

$64\log2-\frac{255}{4}$

✅ Correct Option: 3

Integrating by parts with $\log x$ as the first function gives $\frac{x^4}{4}\log x-\frac{x^4}{16}$ .

At $x=4$ : $64\log4-16$ ; at $x=1$ : $-\frac{1}{16}$ .

The difference is $128\log2-\frac{255}{16}$ , using $64\log4=128\log2$ .

2023: 7 June Shift 1

If $y = \log_3(\log_3 x)$ , then $\frac{dy}{dx}$

2026: 30 May Shift 2

In a company, the cost of producing one item is ₹100, if the item is sold at ₹150 and ₹50,000 is the fixed cost of the company, then the break-even point is:

2023: 11 Aug Shift 1

If $v = \frac{4}{3} \pi r^3$ , at what rate is cubic / unit sec is increasing when $r = 10$ , and $\frac{dr}{dt} = 0.01$ ?

2026: 14 May Shift 2

Match the LIST-I with LIST-II

	LIST-I (Differential equation)		LIST-II (Degree)
A.	$\left(\dfrac{dy}{dx}\right)^2 + \dfrac{1}{\dfrac{dy}{dx}} = 5$	I.	$2$
B.	$\dfrac{d^3y}{dx^3} = \left\{3 + \left(\dfrac{dy}{dx}\right)^2\right\}^{3/2}$	II.	$4$
C.	$\dfrac{d^2y}{dx^2} = \left(\dfrac{dy}{dx}\right)^{5/4}$	III.	$1$
D.	$y\dfrac{d^2x}{dy^2} = y^2 + 1$	IV.	$3$

Choose the correct answer from the options given below:

2023: 23 May Shift 3

The total cost of producing $x$ generators is given by

TC = $x^3 - 60x^2 + 1500x + 2000$ .

The Marginal Cost (MC), when $x = 10$ units is:

2023: 7 June Shift 1

Match List - I with list- II

List-I , Differential equation	List - II , Degree
A. $\left(\frac{dy}{dx}\right)^3 + yx = 0$	2
B. $e^{\frac{dy}{dx}} + y^2 + y'' = 0$	1
C. $Xyy'' + x(y')^2 - yy' = 0$	Not defined
D. $(Y'')^2 + y = 0$	3

Choose the correct option below :

∫14x3log⁡x dx\int_1^4 x^3\log x\, dx∫14​x3logxdx is equal to:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int_1^4 x^3\log x\, dx$ is equal to: