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

∫ x2(x2+1)(x2-2)(x2+3) dx - Mathematics and Statistics

Advertisements

Advertisements

Question

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

Sum

Solution

Let I = $\int \frac{x^{2}}{(x^{2} + 1) (x^{2} - 2) (x^{2} + 3)} d x$

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

= $\frac{A}{x^{2} + 1} + \frac{b}{x^{2} - 2} + \frac{c}{x^{2} + 3}$

∴ x² = A(x² − 2)(x² + 3) + B(x² + 1)(x² + 3) + C(x² + 1)(x² − 2) ........(i)

Putting x² = 2 in (i), we get

2 = B × 3 × 5

∴ B = $\frac{2}{15}$

Putting x² = −3 in (i), we get

−3 = C × (– 2) × (– 5)

∴ C = $\frac{- 3}{10}$

Putting x² = −1 in (i), we get

−1 = A × (–3) × 2

∴ A = $\frac{1}{6}$

∴ $\frac{x^{2}}{(x^{2} + 1) (x^{2} - 2) (x^{2} + 3)} = \frac{\frac{1}{6}}{x^{2} + 1} + \frac{\frac{2}{15}}{x^{2} - 2} + \frac{\frac{- 3}{10}}{x^{2} + 3}$

∴ I = $\int [\frac{1}{6 (x^{2} + 1)} + \frac{2}{15 (x^{2} - 2)} - \frac{3}{10 (x^{2} + 3)}] d x$

= $\frac{1}{6} \int \frac{1}{x^{2} + 1} d x + \frac{2}{15} \int \frac{1}{x^{2} - 2} d x - \frac{3}{10} \int \frac{1}{x^{2} + 3} d x$

= $\frac{1}{6} \int \frac{1}{x^{2} + 1} d x + \frac{2}{15} \int \frac{1}{x^{2} - {(\sqrt{2})}^{2}} d x - \frac{3}{10} \int \frac{1}{x^{2} + {(\sqrt{3})}^{2}} d x$

= $\frac{1}{6} \tan^{- 1} x + \frac{2}{15} \times \frac{1}{2 \times \sqrt{2}} \log | \frac{x - \sqrt{2}}{x + \sqrt{2}} | - \frac{3}{10} \times \frac{1}{\sqrt{3}} \tan^{- 1} (\frac{x}{\sqrt{3}}) + c$

∴ I = $\frac{1}{6} \tan^{- 1} x + \frac{1}{15 \sqrt{2}} \log | \frac{x - \sqrt{2}}{x + \sqrt{2}} | - \frac{\sqrt{3}}{10} \tan^{- 1} (\frac{x}{\sqrt{3}}) + c$

shaalaa.com

Methods of Integration: Integration Using Partial Fractions

Is there an error in this question or solution?

Chapter 2.3: Indefinite Integration - Long Answers III

APPEARS IN

SCERT Maharashtra Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

Chapter 2.3 Indefinite Integration
Long Answers III | Q 8

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

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:

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

Integrate the rational function:

$\frac{2 x}{x^{2} + 3 x + 2}$

Integrate the rational function:

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

Integrate the rational function:

$\frac{3 x + 5}{x^{3} - x^{2} - x + 1}$

Integrate the rational function:

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

Integrate the rational function:

$\frac{5 x}{(x + 1) (x^{2} - 4)}$

Integrate the rational function:

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

Integrate the rational function:

$\frac{1}{x^{4} - 1}$

Integrate the rational function:

$\frac{1}{e^{x} - 1}$ [Hint: Put e^x = t]

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

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

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

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 : $\frac{1}{2 \sin x + \sin 2 x}$

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

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: $\frac{2 x^{2} - 1}{x^{4} + 9 x^{2} + 20}$

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 : $\frac{x^{2}}{\sqrt{1 - x^{6}}}$

Integrate the following w.r.t.x:

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

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

Evaluate: $\int \frac{2 x + 1}{x (x - 1) (x - 4)}$ dx

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

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

$\int x^{2} \sqrt{a^{2} - x^{6}} d x$

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

$\int \frac{\sin x}{\sin 3 x} d x$

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

$\int \sin (\log x) d x$

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

$\int e^{\sin^{- 1_{x}}} [\frac{x + \sqrt{1 - x^{2}}}{\sqrt{1 - x^{2}}}] d x$

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

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

$\int \frac{d x}{2 + 3 \tan x}$

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

Evaluate:

$\int \frac{5 e^{x}}{(e^{x} + 1) (e^{2 x} + 9)} d x$

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

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

$\int \frac{(2 \log x + 3)}{x (3 \log x + 2) [{(\log x)}^{2} + 1]} d x$

$\int \frac{5 (x^{6} + 1)}{x^{2} + 1} d x$ = x⁵ – ______ x³ + 5x + c

$\int \frac{1}{x^{3}} {[{\log x}^{x}]}^{2} d x$ = p(log x)³ + c Then p = ______

Evaluate $\int x^{2} e^{4 x} d x$

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

$\int \frac{1}{4 x^{2} - 20 x + 17} d x$

Verify the following using the concept of integration as an antiderivative

$\int \frac{x^{3} d x}{x + 1} = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \log | x + 1 | + C$

Evaluate the following:

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

Evaluate the following:

$\int \frac{x^{2} d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

If f(x) = $\int (3 x - 1) x (x + 1) {(18 x^{11} + 15 x^{10} - 10 x^{9})}^{\frac{1}{6}} d x$ , where f(0) = 0, is in the form of $\frac{{(18 x^{α} + 15 x^{β} - 10 x^{γ})}^{δ}}{θ}$ , then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)

$\int \frac{1}{{(x^{2} + 1)}^{2}} d x$ = ______.

If $\int \frac{1}{(x^{2} + 4) (x^{2} + 9)} d x = A \tan^{- 1} \frac{x}{2} + B \tan^{- 1} (\frac{x}{3}) + C$ , then A – B = ______.

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.

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

Evaluate:

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

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.