English

CBSECommerce (English Medium) Class 12

Question Papers

Question Papers2580

Textbook Solutions20345

MCQ Online Mock Tests42

Important Solutions22654

Concept Notes & Videos608

If f(x)=∫0πtsin t dt, then f' (x) is ______. - Mathematics

Advertisements

Advertisements

Question

If `f(x) = int_0^pi t sin t dt`, then f' (x) is ______.

Options

cos x + x sin x
x sin x
x cos x
sin x + x cos x

MCQ

Fill in the Blanks

Advertisements

Solution

If `f(x) = int_0^pi t sin t dt`, then f' (x) is x sin x.

Explanation:

f(x) `= int_0^x t sin t dt`

`= [t * (- cos t)]_0^x - int_0^x 1 * (- cos t)` dt

= - x cos x - 0 cos 0 + `(sin t)_0^x"`

= -x cosx + sin x

Hence, f'(x) = -[cos x - x sin x] + cos x

= -cos x + x sin x + cos x

= x sin x

shaalaa.com

Evaluation of Definite Integrals by Substitution

Is there an error in this question or solution?

Chapter 7: Integrals - Exercise 7.9 [Page 340]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 7 Integrals
Exercise 7.9 | Q 10 | Page 340

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Evaluation of Definite Integrals by Substitution

video tutorial00:18:12

RELATED QUESTIONS

Evaluate: `int (1+logx)/(x(2+logx)(3+logx))dx`

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

Evaluate : `int1/(3+5cosx)dx`

If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.

Evaluate :

`int_e^(e^2) dx/(xlogx)`

Evaluate the integral by using substitution.

`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`

Evaluate the integral by using substitution.

`int_(-1)^1 dx/(x^2 + 2x + 5)`

Evaluate the integral by using substitution.

`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`

The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.

Evaluate of the following integral:

\[\int x^\frac{5}{4} dx\]

Evaluate:

\[\int\sqrt{\frac{1 + \cos 2x}{2}}dx\]

Evaluate:

\[\int\sqrt{\frac{1 - \cos 2x}{2}}dx\]

Evaluate :

\[\int\frac{e^{6 \log_e x} - e^{5 \log_e x}}{e^{4 \log_e x} - e^{3 \log_e x}}dx\]

Evaluate:

\[\int\frac{1}{a^x b^x}dx\]

Evaluate:

\[\int\frac{e\log \sqrt{x}}{x}dx\]

\[\int\frac{2x}{\left( 2x + 1 \right)^2} dx\]

Evaluate the following integral:

\[\int\limits_0^4 \left| x - 1 \right| dx\]

Evaluate the following integral:

\[\int\limits_0^4 \left( \left| x \right| + \left| x - 2 \right| + \left| x - 4 \right| \right) dx\]

Evaluate each of the following integral:

\[\int_0^{2\pi} \frac{e^\ sin x}{e^\ sin x + e^{- \ sin x}}dx\]

Evaluate each of the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}}dx\]

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{\tan^2 x}{1 + e^x}dx\]

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{3}}^\frac{\pi}{3} \frac{1}{1 + e^\ tan\ x}dx\]

\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]

Evaluate the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{1}{1 + \cot^\frac{3}{2} x}dx\]

Evaluate the following integral:

\[\int_2^8 \frac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}dx\]

Evaluate the following integral:

\[\int_0^\pi x\sin x \cos^2 xdx\]

Evaluate

\[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\]

Evaluate the following integral:

\[\int_0^{2\pi} \sin^{100} x \cos^{101} xdx\]

Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .

Evaluate: `int_ e^x ((2+sin2x))/cos^2 x dx`

`int_(pi/5)^((3pi)/10) [(tan x)/(tan x + cot x)]`dx = ?

`int_0^1 sin^-1 ((2x)/(1 + x^2))"d"x` = ______.

Each student in a class of 40, studies at least one of the subjects English, Mathematics and Economics. 16 study English, 22 Economics and 26 Mathematics, 5 study English and Economics, 14 Mathematics and Economics and 2 study all the three subjects. The number of students who study English and Mathematics but not Economics is

Find: `int (dx)/sqrt(3 - 2x - x^2)`

Evaluate: `int x/(x^2 + 1)"d"x`

Evaluate:

`int (1 + cosx)/(sin^2x)dx`

If `int x^5 cos (x^6)dx = k sin (x^6) + C`, find the value of k.

Notifications