Choose the correct options from the given alternatives : `int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =

log x – f(x) + c [Hint: `int x^4/(x + x^5)*dx = int((x^4 + 1) - 1)/(x(x^4 + 1))*dx` = `int (1/x - 1/(x + x^5))*dx` = log x – f(x) + c].

Choose the correct options from the given alternatives : ∫1x+x5⋅dx = f(x) + c, then ∫x4x+x5⋅dx = - Mathematics and Statistics

Question

Choose the correct options from the given alternatives :

$\int \frac{1}{x + x^{5}} \cdot d x$ = f(x) + c, then $\int \frac{x^{4}}{x + x^{5}} \cdot d x$ =

Options

log x – f(x) + c
f(x) + log x + c
f(x) – log x + c
$\frac{1}{5} x^{5} f (x) + c$

MCQ

Solution

log x – f(x) + c

[Hint: $\int \frac{x^{4}}{x + x^{5}} \cdot d x = \int \frac{(x^{4} + 1) - 1}{x (x^{4} + 1)} \cdot d x$

= $\int (\frac{1}{x} - \frac{1}{x + x^{5}}) \cdot d x$

= log x – f(x) + c].

shaalaa.com

Methods of Integration: Integration by Parts

Is there an error in this question or solution?

Q 1.02 Q 1.01 Q 1.03

Chapter 3: Indefinite Integration - Miscellaneous Exercise 3 [Page 148]

APPEARS IN

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

Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 1.02 | Page 148

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Methods of Integration: Integration by Parts

video tutorial01:00:57

RELATED QUESTIONS

If $\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\sin^{4} x}{\sin^{4} x + \cos^{4} x} d x$ , then the value of I is:

(A) 0

(B) π

(D) π/4

$\int \frac{1}{x} \log x d x = ... ... ... ... ...$

(A)log(log x)+ c

(B) 1/2 (logx )²+c

(D) log x + c

If u and v are two functions of x then prove that

$\int u v d x = u \int v d x - \int [d \frac{u}{d x} \int v d x] d x$

Hence evaluate, $\int x e^{x} d x$

Evaluate $\int_{0}^{π} e^{2} x . \sin (\frac{π}{4} + x) d x$

Integrate the function in x log 2x.

Integrate the function in x sin^-1 x.

Integrate the function in $\frac{x \cos^{- 1} x}{\sqrt{1 - x^{2}}}$ .

Integrate the function in (x² + 1) log x.

Integrate the function in $\sin^{- 1} (\frac{2 x}{1 + x^{2}})$ .

$\int x^{2} e^{x^{3}} d x$ equals:

Evaluate the following : $\int x^{2} \sin 3 x d x$

Evaluate the following : $\int x^{3} . \tan^{- 1} x . d x$

Evaluate the following : $\int x^{3} . \log x . d x$

Evaluate the following : $\int e^{2 x} . \cos 3 x . d x$

Evaluate the following: $\int x . \sin^{- 1} x . d x$

Evaluate the following : $\int x^{2} \cdot \cos^{- 1} x \cdot d x$

Evaluate the following : $\int \cos \sqrt{x} . d x$

Evaluate the following : $\int \frac{{\sin (\log x)}^{2}}{x} . \log . x . d x$

Evaluate the following : $\int \frac{\log x}{x} . d x$

Integrate the following functions w.r.t. x : $e^{2 x} . \sin 3 x$

Integrate the following functions w.r.t.x:

$e^{- x} \cos 2 x$

Integrate the following functions w.r.t. x : $x^{2} . \sqrt{a^{2} - x^{6}}$

Integrate the following functions w.r.t. x : [2 + cot x – cosec²x]e^x

Integrate the following functions w.r.t. x : $[\frac{x}{{(x + 1)}^{2}}] . e^{x}$

Integrate the following functions w.r.t. x : ${\log (1 + x)}^{(1 + x)}$

Choose the correct options from the given alternatives :

$\int \tan (\sin^{- 1} x) \cdot d x$ =

Choose the correct options from the given alternatives :

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

If f(x) = $\frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}}, g (x) = e^{\sin^{- 1} x}$ , then $\int f (x) \cdot g (x) \cdot d x$ = ______.

Choose the correct options from the given alternatives :

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

Integrate the following w.r.t.x : $\sqrt{x} \sec (x^{\frac{3}{2}}) \cdot \tan (x^{\frac{3}{2}})$

Integrate the following w.r.t.x : log (log x)+(log x)^–2

Integrate the following w.r.t.x : log (x² + 1)

Integrate the following w.r.t.x : sec⁴x cosec²x

Evaluate the following.

$\int x^{2} e^{4x}$ dx

Evaluate the following.

$\int x^{3} e^{x^{2}}$ dx

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

Evaluate: $\int \frac{dx}{{9x}^{2} - 25}$

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

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

Choose the correct alternative:

$\int x^{2} 3^{x^{3}} d x$ =

Choose the correct alternative:

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

$\int {(x + \frac{1}{x})}^{3} d x$ = ______.

$\int \frac{1}{x^{2} - a^{2}} d x$ = ______ + c

$\int \frac{x^{2} + x - 6}{(x - 2) (x - 1)} d x$ = x + ______ + c

Evaluate $\int \frac{1}{x \log x} d x$

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

$\int {[\frac{\log x - 1}{1 + {(\log x)}^{2}}]}^{2}$ dx = ?

∫ log x · (log x + 2) dx = ?

Find $\int_{0}^{1} x (\tan^{- 1} x) d x$

Evaluate the following:

$\int_{0}^{π} x \log \sin x d x$

Find the general solution of the differential equation: $e^{\frac{d y}{d x}} = x^{2}$ .

$\int \frac{1}{\sqrt{x^{2} - a^{2}}} d x$ = ______.

Solve: $\int \sqrt{4 x^{2} + 5} d x$

If $\int \frac{2 e^{5 x} + e^{4 x} - 4 e^{3 x} + 4 e^{2 x} + 2 e^{x}}{(e^{2 x} + 4) {(e^{2 x} - 1)}^{2}} d x = \tan^{- 1} (\frac{e^{x}}{a}) - \frac{1}{b (e^{2 x} - 1)} + C$ , where C is constant of integration, then value of a + b is equal to ______.

If $\int \frac{x + {(\cos^{- 1} 3 x)}^{2}}{\sqrt{1 - 9 x^{2}}} d x = \frac{1}{α} (\sqrt{1 - 9 x^{2}} + {(\cos^{- 1} 3 x)}^{β}) + C$ , where C is constant of integration , then (α + 3β) is equal to ______.

Find $\int \frac{\sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} d x$ .

Evaluate :

$\int \frac{4 x - 6}{{(x^{2} - 3 x + 5)}^{\frac{3}{2}}} d x$

Evaluate the following.

$\int x^{3} e^{x^{2}} d x$

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

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

The integrating factor of $y \log y . \frac{d x}{d y} + x - \log y = 0$ is ______.

Evaluate the following.

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

Evaluate:

$\int {(\log x)}^{2} d x$

Prove that $\int \sqrt{x^{2} - a^{2}} d x = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} \log (x + \sqrt{x^{2} - a^{2}}) + c$

Complete the following activity:

$\int_{0}^{2} \frac{d x}{4 + x - x^{2}}$

= $\int_{0}^{2} \frac{d x}{- x^{2} + □ + □}$

= $\int_{0}^{2} \frac{d x}{- x^{2} + x + \frac{1}{4} - □ + 4}$

= $\int_{0}^{2} \frac{d x}{{(x - \frac{1}{2})}^{2} - {(□)}^{2}}$

= $\frac{1}{\sqrt{17}} \log (\frac{20 + 4 \sqrt{17}}{20 - 4 \sqrt{17}})$

Evaluate the following.

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

Evaluate the following.

$\int x^{3} e^{x^{2}} d x$

If f′(x) = 4x³ − 3x² + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

Evaluate the following.

$\int x^{3} e^{x^{2}} d x$

Evaluate the following.

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

The value of $\int a^{x} . e^{x} d x$ equals

Evaluate:

$\int e^{x} cosec x (1 - \cot x) d x$

Choose the correct options from the given alternatives : ∫1x+x5⋅dx = f(x) + c, then ∫x4x+x5⋅dx = - Mathematics and Statistics

Advertisements

Advertisements

Question

Options

Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Choose the correct options from the given alternatives : ∫1x+x5⋅dx = f(x) + c, then ∫x4x+x5⋅dx = - Mathematics and Statistics

Advertisements

Advertisements

Question

Options

SolutionShow Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Solution