Evaluate the following: eeeed∫e6logx-e5logxe4logx-e3logxdx

प्रश्न

Evaluate the following:

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

बेरीज

उत्तर

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 by Substitution

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 4 Q 3 Q 5

पाठ 7: Integrals - Exercise [पृष्ठ १६३]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12

पाठ 7 Integrals
Exercise | Q 4 | पृष्ठ १६३

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

view
व्हिडिओ ट्यूटोरियल For All Subjects
Evaluation of Definite Integrals by Substitution

video tutorial00:18:12

संबंधित प्रश्‍न

Evaluate: `int1/(xlogxlog(logx))dx`

Evaluate `∫_0^(3/2)|x cosπx|dx`

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Evaluate the integral by using substitution.

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

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

Evaluate the integral by using substitution.

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

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

Evaluate of the following integral:

(i) \[\int x^4 dx\]

Evaluate of the following integral:

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

Evaluate of the following integral:

\[\int\frac{1}{x^5}dx\]

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:

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

Evaluate:

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

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

Evaluate the following definite integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

\[\int\limits_0^2 \left| x^2 - 3x + 2 \right| dx\]

Evaluate the following integral:

\[\int\limits_0^3 \left| 3x - 1 \right| dx\]

Evaluate the following integral:

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

Evaluate the following integral:

\[\int\limits_{- \pi/4}^{\pi/4} \left| \sin x \right| dx\]

Evaluate the following integral:

\[\int\limits_2^8 \left| x - 5 \right| dx\]

Evaluate the following integral:

\[\int\limits_{- \pi/2}^{\pi/2} \left\{ \sin \left| x \right| + \cos \left| x \right| \right\} dx\]

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:

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

Evaluate each of the following integral:

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

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

\[\int_0^\pi \left( \frac{x}{1 + \sin^2 x} + \cos^7 x \right)dx\]

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

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

Evaluate the following:

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

Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.

Evaluate the following: eeeed∫e6logx-e5logxe4logx-e3logxdx

Advertisements

Advertisements

प्रश्न

Advertisements

उत्तर

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Select a course

Evaluate the following: eeeed∫e6logx-e5logxe4logx-e3logxdx

Advertisements

Advertisements

प्रश्न

Advertisements

उत्तरShow Solution

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Select a course

उत्तर