English

CBSECommerce (English Medium) Class 12

Question Papers

Question Papers3000

Textbook Solutions35981

MCQ Online Mock Tests42

Important Solutions23871

Concept Notes & Videos697

Evaluate the following: eeeed∫e6logx-e5logxe4logx-e3logxdx

Advertisements

Advertisements

Question

Evaluate the following:

`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`

Sum

Advertisements

Solution

Let I = `int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`

= `int ("e"^(log x^6) - "e"^(log x^5))/("e"^(logx^4) - "e"^(log x^3)) "d"x` .....[∵ a log b – log b^a]

= `int (x^6 - x^5)/(x^4 - x^3) "d"x` .....[∵ e^logx= x]

= `int (x^3 - x^2)/(x - 1) "d"x`

= `int (x^2(x - 1))/(x - 1) "d"x`

= `int x^2 "d"x`

= `x^3/3 + "C"`

shaalaa.com

Evaluation of Definite Integrals

Is there an error in this question or solution?

Chapter 7: Integrals - Exercise [Page 163]

APPEARS IN

NCERT Exemplar Mathematics Exemplar [English] Class 12

Chapter 7 Integrals
Exercise | Q 4 | Page 163

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Evaluation of Definite Integrals

video tutorial00:18:12

RELATED QUESTIONS

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

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

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

Evaluate: `intsinsqrtx/sqrtxdx`

Evaluate the integral by using substitution.

`int_0^2 xsqrt(x+2)` (Put x + 2 = `t^2`)

Evaluate the integral by using substitution.

`int_0^2 dx/(x + 4 - x^2)`

Evaluate the integral by using substitution.

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

Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`

Evaluate of the following integral:

\[\int\frac{1}{\sqrt[3]{x^2}}dx\]

Evaluate of the following integral:

\[\int 3^{2 \log_3} {}^x dx\]

Evaluate of the following integral:

\[\int \log_x \text{x dx}\]

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:

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

Evaluate:

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

Evaluate the following definite integral:

\[\int_0^1 \frac{1}{\sqrt{\left( x - 1 \right)\left( 2 - x \right)}}dx\]

Evaluate the following integral:

\[\int\limits_{- 6}^6 \left| x + 2 \right| dx\]

Evaluate the following integral:

\[\int\limits_0^{2\pi} \left| \sin x \right| dx\]

Evaluate the following integral:

\[\int\limits_1^4 \left\{ \left| x - 1 \right| + \left| x - 2 \right| + \left| x - 4 \right| \right\} dx\]

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:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx\]

Evaluate each of the following integral:

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

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx\]

Evaluate the following integral:

\[\int_{- \frac{3\pi}{2}}^{- \frac{\pi}{2}} \left\{ \sin^2 \left( 3\pi + x \right) + \left( \pi + x \right)^3 \right\}dx\]

Evaluate :

\[\int\limits_0^{3/2} \left| x \sin \pi x \right|dx\]

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

Find: `int_ (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.

`int_0^3 1/sqrt(3x - x^2)"d"x` = ______.

`int_0^(pi4) sec^4x "d"x` = ______.

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

Evaluate the following:

`int "dt"/sqrt(3"t" - 2"t"^2)`

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

`int_0^1 x^2e^x dx` = ______.

The value of `int_0^1 (x^4(1 - x)^4)/(1 + x^2) dx` is

Evaluate:

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

Which limits are used to evaluate the resulting expression in Method: Resubstitution?

If \[t=\tan^{-1}x\], what is \[dt\]?

What is the value of \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\]?

Notifications