If f(x) = ∫x0 t(sin x-sin t)dt then
If f(x) = ∫x0 t(sin x-sin t)dt then
- $f '''(x)+f ''(x)=\sin x$
- $f '''(x)+f''(x)-f (x)=\cos x$
- $f'''(x)+f '(x)=\cos x-2x \sin x$
- $f '''(x)-f ''(x)=\cos x-2x \sin x$
The Correct Option is C
Solution and Explanation
\(f\left(x\right) = \int^{^x}_{_0}t\left(sin\,x - sin\,t\right)dt\)
\(f\left(x\right) = sinx \int^{^x}_{_0}t\,dt - \int^{^x}_{_0} t \,sin\,tdt\)
\(f'\left(x\right) = \left(sinx\right) x +cosx \int^{^x}_{_0} t \,dt-x\,sinx\)
\(f'\left(x\right) = cosx \int^{^x}_{_0} tdt\) = \(x\cos x\)
\(f''\left(x\right) = \left(cosx\right) x - \left(sinx\right) \int^{^x}_{_0} tdt\)
\(f'''\left(x\right) = x\left(-sinx\right) + cosx-\left(sinx\right)x-\left(cosx\right) \int^{^x}_{_0} tdt\)
\(f'''\left(x\right)+f'\left(x\right) = cosx-2x\,sinx\)
Top Questions on Integrals of Some Particular Functions
- Evaluate the integral: \( \int \sin^5 x \, dx \)
- MHT CET - 2025
- Mathematics
- Integrals of Some Particular Functions
- If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to_______
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If $\int \limits_0^\pi \frac{5^{\cos x}\left(1+\cos x \cos 3 x+\cos ^2 x+\cos ^3 x \cos 3 x\right) d x}{1+5^{\cos x}}=\frac{ k \pi}{16}$, then $k$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If \(\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k}log_e\left \{ \frac{|x-3|^3|x+1|}{(x-2)^4}\right \}+c\), then the value of k is
- WBJEE - 2023
- Mathematics
- Integrals of Some Particular Functions
- The value of $\displaystyle\lim _{n \rightarrow \infty} \frac{1+2-3+4+5-6+\ldots +(3 n-2)+(3 n-1)-3 n}{\sqrt{2 n^4+4 n+3-} \sqrt{n^4+5 n+4}}$ is :
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
Questions Asked in JEE Main exam
Consider the following sequence of reactions :
Molar mass of the product formed (A) is ______ g mol\(^{-1}\).- JEE Main - 2025
- Organic Chemistry
- A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:
- JEE Main - 2025
- electrostatic potential and capacitance
- Three infinitely long wires with linear charge density \( \lambda \) are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?
- JEE Main - 2025
- electrostatic potential and capacitance
Two capacitors \( C_1 \) and \( C_2 \) are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are \( U_1 \) and \( U_2 \), respectively. Which of the given statements is true?
- JEE Main - 2025
- electrostatic potential and capacitance
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.
Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below:- JEE Main - 2025
- electrostatic potential and capacitance
Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
