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IntegrationQuestion and Answers: Page 322

Question Number 21721 Answers: 1 Comments: 0

∫_0 ^(π/2) sin^2 xcos^3 xdx

$\int_{0}^{π / 2} {s i n}^{2} {x c o s}^{3} x d x$

Question Number 21720 Answers: 1 Comments: 0

∫sin^5 θdθ

$\int {s i n}^{5} θ d θ$

Question Number 21707 Answers: 0 Comments: 3

∫_(π/6) ^(π/3) (1/2)cot^2 2θdθ

$\int_{π / 6}^{π / 3} \frac{1}{2} {c o t}^{2} 2 θ d θ$

Question Number 21702 Answers: 0 Comments: 1

∫_(π/6) ^(π/3) 1/2cot^2 2θdθ

$\int_{π / 6}^{π / 3} 1 / 2 {c o t}^{2} 2 θ d θ$

Question Number 21701 Answers: 2 Comments: 1

∫2cot^2 2t

$\int 2 {c o t}^{2} 2 t$

Question Number 21662 Answers: 1 Comments: 0

Question Number 21679 Answers: 1 Comments: 0

∫((secθ dθ)/(1−secθ))

$\int \frac{\sec θ d θ}{1 - \sec θ}$

Question Number 21680 Answers: 1 Comments: 0

∫(√(secθ)) dθ

$\int \sqrt{\sec θ} d θ$

Question Number 21634 Answers: 1 Comments: 0

∫3tan2t

$\int 3 t a n 2 t$

Question Number 21708 Answers: 1 Comments: 0

∫_0 ^(0.5) 2tan^2 2tdt

$\int_{0}^{0.5} 2 {t a n}^{2} 2 t d t$

Question Number 21591 Answers: 1 Comments: 1

5cos^5 tsin t

$5 \cos^{5} t \sin t$

Question Number 21595 Answers: 1 Comments: 1

3sec^2 3xtan3x

${3 \sec}^{2} 3 x t a n 3 x$

Question Number 21582 Answers: 1 Comments: 0

∫_(π/2) ^(π/4) (3x+7)

$\int_{π / 2}^{π / 4} (3 x + 7)$

Question Number 21450 Answers: 0 Comments: 0

Question Number 21390 Answers: 0 Comments: 0

∫_0 ^1 ((x^7 − 1)/(ln x)) dx

$\underset{0}{\int^{1}} \frac{x^{7} - 1}{\ln x} d x$

Question Number 21212 Answers: 0 Comments: 0

Question Number 21205 Answers: 0 Comments: 1

Question Number 21077 Answers: 1 Comments: 0

integrate with respect to x ∫(dx/(9x^2 +6x+10))

$i n t e g r a t e w i t h r e s p e c t t o x$ $\int \frac{d x}{9 x^{2} + 6 x + 10}$

Question Number 21076 Answers: 1 Comments: 0

integrate with respect to x ∫x^(sinx)

$i n t e g r a t e w i t h r e s p e c t t o x$ $\int x^{s i n x}$

Question Number 20939 Answers: 0 Comments: 0

Demostration of the volume of an sphere V=((4πr^3 )/3) x^2 +y^2 +z^2 =r^2 We divide the sphere in 8 parts. So the volume of a part is ∫_0 ^( r) ∫_0 ^( (√(r^2 −x^2 ))) (√(r^2 −x^2 −y^2 ))∂y∂x Lets asumme a^2 =r^2 −x^2 ∫_0 ^( r) ∫_0 ^( a) (√(a^2 −y^2 ))∂y∂x ∫_0 ^( r) a∫_0 ^( a) (√(1−((y/a))^2 ))∂y∂x Lets assume (y/a)=sinθ⇒(∂y/a)=cosθ∂θ ∫(√(1−sin^2 θ))acosθ∂θ ∫acos^2 θ∂θ a((θ/2)−((sin2θ)/4)) a(((arcsin((y/a)))/2)−((y(√(a^2 −y^2 )))/(2a^2 ))) ∫_0 ^( r) a^2 (((arcsin((y/a)))/2)−((y(√(a^2 −y^2 )))/(2a^2 )))∣_0 ^a ∂x ∫_0 ^( r) (((a^2 arcsin((y/a))−y(√(a^2 −y^2 )))/2))∣_0 ^a ∂x ∫_0 ^( r) ((πa^2 )/4)∂x ∫_0 ^( r) ((π(r^2 −x^2 ))/4)∂x (((6πr^2 x−2πx^3 )/(24)))∣_0 ^r ((πr^3 )/6)=1/8Volume of the sphrere so... V=((4πr^3 )/3)

$Demostration of the volume of an sphere V = \frac{4 π r^{3}}{3}$ $x^{2} + y^{2} + z^{2} = r^{2} We divide the sphere in 8 parts . So the volume of a part is$ $\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \sqrt{r^{2} - x^{2} - y^{2}} \partial y \partial x Lets asumme a^{2} = r^{2} - x^{2}$ $\int_{0}^{r} \int_{0}^{a} \sqrt{a^{2} - y^{2}} \partial y \partial x$ $\int_{0}^{r} a \int_{0}^{a} \sqrt{1 - {(\frac{y}{a})}^{2}} \partial y \partial x Lets assume \frac{y}{a} = \sin θ \Rightarrow \frac{\partial y}{a} = \cos θ \partial θ$ $\int \sqrt{1 - \sin^{2} θ} acos θ \partial θ$ $\int {acos}^{2} θ \partial θ$ $a (\frac{θ}{2} - \frac{\sin 2 θ}{4})$ $a (\frac{\arcsin (\frac{y}{a})}{2} - \frac{y \sqrt{a^{2} - y^{2}}}{{2 a}^{2}})$ $\int_{0}^{r} a^{2} (\frac{\arcsin (\frac{y}{a})}{2} - \frac{y \sqrt{a^{2} - y^{2}}}{{2 a}^{2}}) ∣_{0}^{a} \partial x$ $\int_{0}^{r} (\frac{a^{2} \arcsin (\frac{y}{a}) - y \sqrt{a^{2} - y^{2}}}{2}) ∣_{0}^{a} \partial x$ $\int_{0}^{r} \frac{π a^{2}}{4} \partial x$ $\int_{0}^{r} \frac{π (r^{2} - x^{2})}{4} \partial x$ $(\frac{6 π r^{2} x - 2 π x^{3}}{24}) ∣_{0}^{r}$ $\frac{π r^{3}}{6} = 1 / 8 Volume of the sphrere so . . .$ $V = \frac{4 π r^{3}}{3}$

Question Number 20908 Answers: 1 Comments: 0

integrate with respect to x ∫(((2x+1)/(x^2 +4x+8)))dx

$i n t e g r a t e w i t h r e s p e c t t o x$ $\int (\frac{2 x + 1}{x^{2} + 4 x + 8}) d x$

Question Number 20905 Answers: 1 Comments: 1

∫e^x^2 dx

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

Question Number 20873 Answers: 1 Comments: 0

∫_1 ^5 (e^x /x^2 ) dx

$\int_{1}^{5} \frac{e^{x}}{x^{2}} d x$

Question Number 20857 Answers: 0 Comments: 0

∫ (√(sin x)) + (√(cos x)) dx

$\int \sqrt{\sin x} + \sqrt{\cos x} d x$

Question Number 20733 Answers: 1 Comments: 0

Find the volume of the solid of revolution obtained by revolving area bounded by x = 4 + 6y − 2y^2 , x = −4, x = 0 about the y−axis

$Find the volume of the solid of revolution$ $obtained by revolving area bounded by$ $x = 4 + 6 y - 2 y^{2}, x = - 4, x = 0 about$ $the y - axis$

Question Number 20686 Answers: 2 Comments: 1

∫(tanx)^(1/3) dx

$\int {(t a n x)}^{1 / 3} d x$

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