Choose the correct options from the given alternatives : ∫tan(sin-1x)⋅dx = - Mathematics and Statistics

प्रश्न

Choose the correct options from the given alternatives :

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

विकल्प

${(1 - x^{2})}^{- \frac{1}{2}} + c$
${(1 - x^{2})}^{\frac{1}{2}} + c$
$\frac{\tan^{m} x}{\sqrt{1 - x^{2}}} + c$
$- \sqrt{1 - x^{2}} + c$

MCQ

उत्तर

$- \sqrt{1 - x^{2}} + c$

$[Hint : \sin^{- 1} x = \tan^{- 1} (\frac{x}{\sqrt{1 - x^{2}}})] .$

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Methods of Integration: Integration by Parts

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Q 1.05 Q 1.04 Q 1.06

अध्याय 3: Indefinite Integration - Miscellaneous Exercise 3 [पृष्ठ १४८]

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

अध्याय 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 1.05 | पृष्ठ १४८

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Methods of Integration: Integration by Parts

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Prove that:

$\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} (\frac{x}{a}) + c$

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

(A)log(log x)+ c

(B) 1/2 (logx )²+c

(D) log x + c

Integrate the function in x sin 3x.

Integrate the function in x sin^-1 x.

Integrate the function in x tan^-1 x.

Integrate the function in (sin^-1x)².

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

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

$\int e^{x} \sec x (1 + \tan x) d x$ equals:

Evaluate the following : $\int x^{2} . \log x . d x$

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

Evaluate the following : $\int x^{2} \tan^{- 1} x . d x$

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

Evaluate the following:

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

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

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

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

Integrate the following functions w.r.t. x : $\sqrt{4^{x} (4^{x} + 4)}$

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

Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]

Choose the correct options from the given alternatives :

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

Choose the correct options from the given alternatives :

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Choose the correct options from the given alternatives :

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

Integrate the following with respect to the respective variable : cos 3x cos 2x cos x

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

Solve the following differential equation.

(x² − yx² ) dy + (y² + xy²) dx = 0

Evaluate the following.

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

Evaluate the following.

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

Evaluate the following.

$\int e^{x} [{(\log x)}^{2} + \frac{2 \log x}{x}]$ dx

Evaluate the following.

$\int \frac{\log x}{{(1 + \log x)}^{2}}$ dx

Choose the correct alternative from the following.

$\int \frac{(x^{3} + 3 x^{2} + 3 x + 1)}{{(x + 1)}^{5}} dx$ =

Evaluate: Find the primitive of $\frac{1}{1 + e^{x}}$

Evaluate: $\int \frac{dx}{\sqrt{4 x^{2} - 5}}$

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

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

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

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

$\int \sin 4 x \cos 3 x d x$

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

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

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

$\int_{0}^{a} \sqrt{\frac{x}{a - x}} dx$ = ____________.

$\int \log x \cdot {[\log (e x)]}^{- 2}$ dx = ?

$\int e^{x} \int [\frac{2 - \sin 2 x}{1 - \cos 2 x}]$ dx = ______.

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$\int_{0}^{1} x \log (1 + 2 x) d x$

State whether the following statement is true or false.

If $\int \frac{4 e^{x} - 25}{2 e^{x} - 5}$ dx = Ax – 3 log |2e^x – 5| + c, where c is the constant of integration, then A = 5.

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

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

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

$\int e^{x} [\frac{2 + \sin 2 x}{1 + \cos 2 x}] d x$ = ______.

$\int_{0}^{1} x \tan^{- 1} x d x$ = ______.

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

Solution of the equation $x \frac{d y}{d x} = y \log y$ is ______

Solve the differential equation (x²+ y²) dx - 2xy dy = 0 by completing the following activity.

Solution: (x²+ y²) dx - 2xy dy = 0

∴ $\frac{d y}{d x} = \frac{x^{2} + y^{2}}{2 x y}$ ...(1)

Puty = vx

∴ $\frac{d y}{d x} = □$

∴ equation (1) becomes

$x \frac{d v}{d x} = □$

∴ $□ d v = \frac{d x}{x}$

On integrating, we get

$\int \frac{2 v}{1 - v^{2}} d v = \int \frac{d x}{x}$

∴ $- \log | 1 - v^{2} | = \log | x | + c_{1}$

∴ $\log | x | + \log | 1 - v^{2} | = \log c ... [where - c_{1} = \log c]$

∴ x(1 - v²) = c

By putting the value of v, the general solution of the D.E. is $□$ = cx

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

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

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

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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.

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

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

Evaluate the following.

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

Evaluate the following.

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

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

Choose the correct options from the given alternatives : ∫tan(sin-1x)⋅dx = - Mathematics and Statistics

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Choose the correct options from the given alternatives : ∫tan(sin-1x)⋅dx = - Mathematics and Statistics

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