English

Maharashtra State BoardHSC Science (General) 12th Standard Board Exam

Question Papers

Question Papers306

Textbook Solutions13141

MCQ Online Mock Tests73

Important Solutions7886

Concept Notes & Videos461

Integrate the following w.r.t. x : 2x4-3x-x2 - Mathematics and Statistics

Advertisements

Advertisements

Question

Integrate the following w.r.t. x : $\frac{2 x}{4 - 3 x - x^{2}}$

Sum

Solution

Let I = $\int \frac{2 x}{4 - 3 x - x^{2}} . d x$

Let $\frac{2 x}{4 - 3 x - x^{2}}$

= $\frac{2 x}{(4 + x) (1 - x)}$

= $\frac{A}{4 + x} + \frac{B}{1 - x}$

∴ 2x = A(1 – x) + B(4 + x)
Put 4 + x = 0, i.e. x = – 4, we get
– 8 = A(5) + B(0)

∴ A = $- \frac{8}{5}$
Put 1 – x = 0, i.e x = 1, we
2 = A(0) + B(5)

∴ B = $\frac{2}{5}$

∴ $\frac{2 x}{4 - 3 x - x^{2}} = \frac{(- \frac{8}{5})}{4 + x} + \frac{(\frac{2}{5})}{1 - x}$

∴ I = $\int [\frac{\frac{- 8}{5}}{4 + x} + \frac{(\frac{2}{5})}{1 - x}] . d x$

= $- \frac{8}{5} \int \frac{1}{4 + x} . d x + \frac{2}{5} \int \frac{1}{1 - x} . d x$

= $- \frac{8}{5} \log | 4 + x | + \frac{2}{5} . \frac{\log | 1 - x |}{- 1} + c$

= $- \frac{8}{5} \log | 4 + x | - \frac{2}{5} \log | 1 - x | + c$ .

shaalaa.com

Methods of Integration: Integration Using Partial Fractions

Is there an error in this question or solution?

Q 1.04 Q 1.03 Q 1.05

Chapter 3: Indefinite Integration - Exercise 3.4 [Page 145]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board

Chapter 3 Indefinite Integration
Exercise 3.4 | Q 1.04 | Page 145

Video TutorialsVIEW ALL [2]

view
Video Tutorials For All Subjects
Methods of Integration: Integration Using Partial Fractions

video tutorial00:18:25
Methods of Integration: Integration Using Partial Fractions

video tutorial00:42:15

RELATED QUESTIONS

Evaluate : $\int \frac{x^{2}}{(x^{2} + 2) (2 x^{2} + 1)} d x$

Evaluate:

$\int \frac{x^{2}}{x^{4} + x^{2} - 2} d x$

Evaluate: $\int \frac{8}{(x + 2) (x^{2} + 4)} d x$

Integrate the rational function:

$\frac{x}{(x + 1) (x + 2)}$

Integrate the rational function:

$\frac{1}{x^{2} - 9}$

Integrate the rational function:

$\frac{3 x - 1}{(x - 1) (x - 2) (x - 3)}$

Integrate the rational function:

$\frac{1 - x^{2}}{x (1 - 2 x)}$

Integrate the rational function:

$\frac{x}{(x^{2} + 1) (x - 1)}$

Integrate the rational function:

$\frac{2 x - 3}{(x^{2} - 1) (2 x + 3)}$

Integrate the rational function:

$\frac{3 x - 1}{{(x + 2)}^{2}}$

Integrate the rational function:

$\frac{2 x}{(x^{2} + 1) (x^{2} + 3)}$

$\int \frac{d x}{x (x^{2} + 1)}$ equals:

Find $\int \frac{e^{x} d x}{{(e^{x} - 1)}^{2} (e^{x} + 2)}$

Find $\int \frac{2 \cos x}{(1 - \sin x) (1 + \sin^{2} x)} d x$

Find :

$\int \frac{\sin (x - a)}{\sin (x + a)} d x$

Integrate the following w.r.t. x : $\frac{x^{2} + 2}{(x - 1) (x + 2) (x + 3)}$

Integrate the following w.r.t. x : $\frac{x^{2} + x - 1}{x^{2} + x - 6}$

Integrate the following w.r.t. x : $\frac{12 x^{2} - 2 x - 9}{(4 x^{2} - 1) (x + 3)}$

Integrate the following w.r.t. x : $\frac{2^{x}}{4^{x} - 3 \cdot 2^{x} - 4}$

Integrate the following w.r.t. x : $\frac{5 x^{2} + 20 x + 6}{x^{3} + 2 x^{2} + x}$

Integrate the following w.r.t. x : $\frac{1}{x (1 + 4 x^{3} + 3 x^{6})}$

Integrate the following w.r.t. x : $\frac{1}{x^{3} - 1}$

Integrate the following w.r.t. x : $\frac{(3 \sin - 2) \cdot \cos x}{5 - 4 \sin x - \cos^{2} x}$

Integrate the following w.r.t. x : $\frac{5 \cdot e^{x}}{(e^{x} + 1) (e^{2 x} + 9)}$

Integrate the following w.r.t. x: $\frac{x^{2} + 3}{(x^{2} - 1) (x^{2} - 2)}$

Integrate the following with respect to the respective variable : $\cot^{- 1} (\frac{1 + \sin x}{\cos x})$

Integrate the following w.r.t.x : $\frac{1}{(1 - \cos 4 x) (3 - \cot 2 x)}$

Integrate the following w.r.t.x : $\frac{1}{\sin x + \sin 2 x}$

Integrate the following w.r.t.x : $\sec^{2} x \sqrt{7 + 2 \tan x - \tan^{2} x}$

Integrate the following w.r.t.x : $\frac{\sqrt{\tan x}}{\sin x \cdot \cos x}$

Evaluate: $\int \frac{x^{2} + x - 1}{x^{2} + x - 6}$ dx

Evaluate: $\int \frac{1}{x (x^{5} + 1)}$ dx

Evaluate: $\int \frac{1}{x (x^{n} + 1)}$ dx

$\int \frac{dx}{(x - 8) (x + 7)}$ =

State whether the following statement is True or False.

If $\int \frac{(x - 1) dx}{(x + 1) (x - 2)}$ = A log |x + 1| + B log |x - 2| + c, then A + B = 1.

$\int e^{3 \log x} {(x^{4} + 1)}^{- 1} d x$

If f'(x) = $x - \frac{3}{x^{3}}$ , f(1) = $\frac{11}{2}$ find f(x)

$\int \frac{(x^{2} + 2)}{x^{2} + 1} a^{x + \tan^{- 1_{x}}} d x$

$\int \frac{1}{2 + \cos x - \sin x} d x$

$\int \sec^{3} x d x$

$\int \sec^{2} x \sqrt{\tan^{2} x + \tan x - 7} d x$

$\int \frac{x^{2} + x - 1}{x^{2} + x - 6} d x$

$\int \frac{6 x^{3} + 5 x^{2} - 7}{3 x^{2} - 2 x - 1} d x$

$\int \frac{3 x + 4}{\sqrt{2 x^{2} + 2 x + 1}} d x$

$\int x^{3} \tan^{- 1} x d x$

$\int \frac{x + \sin x}{1 - \cos x} d x$

Choose the correct alternative:

$\int \sqrt{1 + x} d x$ =

Choose the correct alternative:

$\int \frac{x + 2}{2 x^{2} + 6 x + 5} d x = p \int \frac{4 x + 6}{2 x^{2} + 6 x + 5} d x + \frac{1}{2} \int \frac{1}{2 x^{2} + 6 x + 5} d x$ , then p = ?

State whether the following statement is True or False:

For $\int \frac{x - 1}{{(x + 1)}^{3}} e^{x} d x$ = e^xf(x) + c, f(x) = (x + 1)²

If $\int \frac{\sin 2 x}{\sin 5 x \sin 3 x} d x = \frac{1}{3} \log | \sin 3 x | - \frac{1}{5} \log | f (x) | + c$ , then f(x) = ______

Evaluate the following:

$\int \frac{x^{2}}{1 - x^{4}} d x$ put x² = t

Evaluate the following:

$\int \frac{x^{2} d x}{x^{4} - x^{2} - 12}$

Evaluate the following:

$\int \frac{2 x - 1}{(x - 1) (x + 2) (x - 3)} d x$

Evaluate: $\int \frac{d x}{2 + \cos x - \sin x}$

Find: $\int \frac{x^{2}}{(x^{2} + 1) (3 x^{2} + 4)} d x$

Evaluate: $\int_{- 2}^{1} \sqrt{5 - 4 x - x^{2}} d x$

Let g : (0, ∞) $\to$ R be a differentiable function such that $\int (\frac{x (\cos x - \sin x)}{e^{x} + 1} + \frac{g (x) (e^{x} + 1 - x e^{x})}{{(e^{x} + 1)}^{2}}) d x = \frac{x g (x)}{e^{x} + 1} + c$ , for all x > 0, where c is an arbitrary constant. Then ______.

If $\int \frac{d x}{\sqrt{16 - 9 x^{2}}}$ = A sin^–1 (Bx) + C then A + B = ______.

Evaluate: $\int \frac{2 x^{2} - 3}{(x^{2} - 5) (x^{2} + 4)} d x$

Find: $\int \frac{x^{4}}{(x - 1) (x^{2} + 1)} d x$ .

Find : $\int \frac{2 x^{2} + 3}{x^{2} (x^{2} + 9)} d x; x \neq 0$ .

Evaluate $\int \frac{5 x^{2} - 6 x + 3}{2 x - 3} d x$

Evaluate:

$\int \frac{x}{(x + 2) {(x - 1)}^{2}} d x$

Evaluate:

$\int \frac{x + 7}{x^{2} + 4 x + 7} d x$

Evaluate:

$\int \frac{2 x^{3} - 1}{x^{4} + x} d x$

Notifications

Our website is made possible by ad-free subscriptions or displaying online advertisements to our visitors.
If you don't like ads you can support us by buying an ad-free subscription or please consider supporting us by disabling your ad blocker. Thank you.