English

CBSECommerce (English Medium) Class 12

Question Papers

Question Papers2890

Textbook Solutions20276

MCQ Online Mock Tests42

Important Solutions23871

Concept Notes & Videos601

3 ∫ 1 ( X 2 + 3 X ) D X

Advertisements

Advertisements

Question

\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]

Sum

Advertisements

Solution

\[\text{Here }a = 1, b = 3, f\left( x \right) = x^2 + 3x, h = \frac{3 - 1}{n} = \frac{2}{n}\]

Therefore,

\[ \int_1^3 \left( x^2 + 3x \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) + . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]

\[ = \lim_{h \to 0} h\left[ f\left( 1 \right) + f\left( 1 + h \right) + . . . . . . . . . . + f\left( 1 + \left( n - 1 \right)h \right) \right]\]

\[ = \lim_{h \to 0} h\left[ 1 + 3 + \left( 1 + h \right)^2 + 3\left( 1 + h \right) + \left( 1 + 2h \right)^2 + 3\left( 1 + 2h \right) + . . . . . . . . . + \left( \left( n - 1 \right)h \right)^2 + 3\left( \left( n - 1 \right)h \right) \right]\]

\[ = \lim_{h \to 0} h\left[ n + h^2 \left( 1^2 + 2^2 + . . . . . . . . . . . . . . \left( n - 1 \right)^2 \right) + 2h\left( 1 + 2 + . . . . . . . . . . . . \left( n - 1 \right) \right) + 3n + 3h\left( 1 + 2 + . . . . . . . . . . . . \left( n - 1 \right) \right) \right]\]

\[ = \lim_{h \to 0} h\left[ 4n + h^2 \frac{n\left( n - 1 \right)\left( 2n - 1 \right)}{6} + 5h\frac{n\left( n - 1 \right)}{2} \right]\]

\[ = \lim_{n \to 0 } \left[ 8 + \frac{4}{3}\left( 1 - \frac{1}{n} \right)\left( 2 - \frac{1}{n} \right) + 10\left( 1 - \frac{1}{n} \right) \right]\]

\[ = 8 + \frac{8}{3} + 10\]

\[ = \frac{62}{3}\]

shaalaa.com

Definite Integrals

Is there an error in this question or solution?

Chapter 19: Definite Integrals - Revision Exercise [Page 123]

APPEARS IN

R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12

Chapter 19 Definite Integrals
Revision Exercise | Q 67 | Page 123

RELATED QUESTIONS

\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]

\[\int\limits_0^{\pi/6} \cos x \cos 2x\ dx\]

\[\int\limits_0^{\pi/2} \sqrt{1 + \cos x}\ dx\]

Evaluate the following definite integrals:

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

\[\int\limits_e^{e^2} \left\{ \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right\} dx\]

\[\int\limits_0^1 \left( x e^{2x} + \sin\frac{\ pix}{2} \right) dx\]

\[\int_0^\pi e^{2x} \cdot \sin\left( \frac{\pi}{4} + x \right) dx\]

\[\int_0^1 x\log\left( 1 + 2x \right)dx\]

\[\int_0^1 \frac{1}{1 + 2x + 2 x^2 + 2 x^3 + x^4}dx\]

\[\int\limits_1^3 \frac{\cos \left( \log x \right)}{x} dx\]

\[\int\limits_0^{\pi/2} \sqrt{\sin \phi} \cos^5 \phi\ d\phi\]

\[\int\limits_0^{\pi/2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d\theta\]

\[\int\limits_0^1 \frac{\sqrt{\tan^{- 1} x}}{1 + x^2} dx\]

\[\int\limits_0^{\pi/4} \left( \sqrt{\tan}x + \sqrt{\cot}x \right) dx\]

\[\int\limits_0^1 \frac{1 - x^2}{\left( 1 + x^2 \right)^2} dx\]

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

\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{- \frac{\pi}{2}}{\sqrt{\cos x \sin^2 x}}dx\]

\[\int\limits_0^5 \frac{\sqrt[4]{x + 4}}{\sqrt[4]{x + 4} + \sqrt[4]{9 - x}} dx\]

\[\int\limits_0^7 \frac{\sqrt[3]{x}}{\sqrt[3]{x} + \sqrt[3]{7} - x} dx\]

\[\int\limits_{- \pi/4}^{\pi/4} \sin^2 x\ dx\]

If f is an integrable function, show that

\[\int\limits_{- a}^a x f\left( x^2 \right) dx = 0\]

\[\int\limits_0^2 \left( x^2 + 1 \right) dx\]

\[\int\limits_0^3 \frac{1}{x^2 + 9} dx .\]

The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is

\[\int\limits_0^3 \frac{3x + 1}{x^2 + 9} dx =\]

\[\int\limits_{- \pi/2}^{\pi/2} \sin\left| x \right| dx\] is equal to

\[\int\limits_0^\infty \log\left( x + \frac{1}{x} \right) \frac{1}{1 + x^2} dx =\]

\[\int\limits_0^{2a} f\left( x \right) dx\] is equal to

The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is

Evaluate : \[\int e^{2x} \cdot \sin \left( 3x + 1 \right) dx\] .

\[\int\limits_0^1 \cos^{- 1} x dx\]

Evaluate the following integrals :-

\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]

\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]

\[\int\limits_0^\pi x \sin x \cos^4 x dx\]

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

\[\int\limits_0^\pi \cos 2x \log \sin x dx\]

\[\int\limits_0^1 \cot^{- 1} \left( 1 - x + x^2 \right) dx\]

Evaluate the following using properties of definite integral:

`int_(- pi/4)^(pi/4) x^3 cos^3 x "d"x`

Choose the correct alternative:

Γ(n) is

`int "e"^x ((1 - x)/(1 + x^2))^2 "d"x` is equal to ______.

Notifications