हिंदी

महाराष्ट्र स्टेट बोर्डएचएससी विज्ञान (सामान्य) १२ वीं कक्षा

प्रश्न पत्र

प्रश्न पत्र३०६

पाठ्यपुस्तक उत्तर१३१४१

MCQ ऑनलाइन अभ्यास परीक्षा७३

महत्वपूर्ण प्रश्न एवं उत्तर७८८६

अवधारणा स्पष्टीकरण और वीडियो४६१

समय सारणी२८

पाठ्यक्रम

∫1x(x3-1)dx - Mathematics and Statistics

Advertisements

Advertisements

प्रश्न

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

योग

उत्तर

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

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

= $\int \frac{1}{x^{4} (1 - \frac{1}{x^{3}})} d x$

Put $1 - \frac{1}{x^{3}}$ = t

Differentiating w.r.t.x, we get

$\frac{3}{x^{4}} d x$ = dt

∴ $\frac{1}{x^{4}} d x = \frac{1}{3} dt$

∴I = $\frac{1}{3} \int \frac{dt}{t}$

= $\frac{1}{3} \log | t | + c$

= $\frac{1}{3} \log | 1 - \frac{1}{x^{3}} | + c$

∴ I = $\frac{1}{3} \log | \frac{x^{3} - 1}{x^{3}} | + c$

shaalaa.com

Methods of Integration: Integration Using Partial Fractions

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

अध्याय 2.3: Indefinite Integration - Short Answers II

APPEARS IN

एससीईआरटी महाराष्ट्र Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

अध्याय 2.3 Indefinite Integration
Short Answers II | Q 1

वीडियो ट्यूटोरियलVIEW ALL [2]

view
वीडियो ट्यूटोरियल 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

संबंधित प्रश्न

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

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

Evaluate:

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

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:

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

Integrate the rational function:

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

Integrate the rational function:

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

Integrate the rational function:

$\frac{1}{x (x^{n} + 1)}$ [Hint: multiply numerator and denominator by xⁿ^{− 1} and put xⁿ = t]

Integrate the rational function:

$\frac{(x^{2} + 1) (x^{2} + 2)}{(x^{2} + 3) (x^{2} + 4)}$

Integrate the rational function:

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

Integrate the following w.r.t. x : $\frac{12 x + 3}{6 x^{2} + 13 x - 63}$

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

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

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

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

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 : $\frac{\cos 7 x - \cos 8 x}{1 + 2 \cos 5 x}$

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

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

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

Evaluate: $\int \frac{3x - 2}{{(x + 1)}^{2} (x + 3)}$ dx

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

Evaluate: $\int \frac{3x - 1}{{2x}^{2} - x - 1}$ dx

Evaluate: $\int \frac{1 + \log x}{x (3 + \log x) (2 + 3 \log x)}$ dx

$\int \frac{2 x - 7}{\sqrt{4 x - 1}} d x$

$\int \frac{x^{7}}{{(1 + x^{4})}^{2}} d x$

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

$\int \frac{7 + 4 x + 5 x^{2}}{{(2 x + 3)}^{\frac{3}{2}}} d x$

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

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

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

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

$\int x \sin 2 x \cos 5 x d x$

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

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

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

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 = ?

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

If f'(x) = $\frac{1}{x} + x$ and f(1) = $\frac{5}{2}$ , then f(x) = log x + $\frac{x^{2}}{2}$ + ______ + c

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

Evaluate $\int x \log x d x$

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

$\int \frac{3 e^{2 t} + 5}{4 e^{2 t} - 5} dt$

If $\int \sqrt{\frac{x - 7}{x - 9}} d x = A \sqrt{x^{2} - 16 x + 63} + \log | x - 8 + \sqrt{x^{2} - 16 x + 63} | + c$ , then A = ______

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 the following:

$\int e^{- 3 x} \cos^{3} x d x$

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

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

If $\int \sqrt{\frac{x - 5}{x - 7}} d x = A \sqrt{x^{2} - 12 x + 35} + \log | x | - 6 + \sqrt{x^{2} - 12 x + 35} + C ∣$ , then A = ______.

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:

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

Evaluate.

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

Evaluate:

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

Notifications

English हिंदी मराठी

Login

Create free account

Course

एचएससी विज्ञान (सामान्य) १२ वीं कक्षा महाराष्ट्र स्टेट बोर्ड

Change mode
Log out

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.