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IntegrationQuestion and Answers: Page 284

Question Number 36441 Answers: 1 Comments: 1

find ∫ (e^(tanx) /(cos^2 x))dx

$f i n d \int \frac{e^{t a n x}}{{c o s}^{2} x} d x$

Question Number 36440 Answers: 0 Comments: 0

find ∫ (dx/((3+x^2 )^(1/3) ))

$f i n d \int \frac{d x}{{(3 + x^{2})}^{\frac{1}{3}}}$

Question Number 36439 Answers: 1 Comments: 1

calculate ∫_0 ^π ((sin(2x))/(2 +cosx))dx

$c a l c u l a t e \int_{0}^{π} \frac{s i n (2 x)}{2 + c o s x} d x$

Question Number 36438 Answers: 1 Comments: 3

let F(x) =∫_x ^(1/x) ((arctan(t))/t)dt 1) calculate (dF/dx)(x) 2) find F(x).

$l e t F (x) = \int_{x}^{\frac{1}{x}} \frac{a r c t a n (t)}{t} d t$ $1) c a l c u l a t e \frac{d F}{d x} (x)$ $2) f i n d F (x) .$

Question Number 36436 Answers: 2 Comments: 0

find ∫ ((sinx)/(1+cos^3 x))dx

$f i n d \int \frac{s i n x}{1 + {c o s}^{3} x} d x$

Question Number 36435 Answers: 0 Comments: 4

find the value of h(t)=∫_0 ^1 ln(1+tx^2 ) with ∣t∣≤1 2) calculate ∫_0 ^1 ln(1+x^2 )dx 3) calculate ∫_0 ^1 ln(1−x^2 )dx

$f i n d t h e v a l u e o f h (t) = \int_{0}^{1} l n (1 + {t x}^{2}) w i t h ∣ t ∣⩽ 1$ $2) c a l c u l a t e \int_{0}^{1} l n (1 + x^{2}) d x$ $3) c a l c u l a t e \int_{0}^{1} l n (1 - x^{2}) d x$

Question Number 36434 Answers: 1 Comments: 1

find g(x) =∫_0 ^x (e^(−t) /(√(1+t^2 )))dt.

$f i n d g (x) = \int_{0}^{x} \frac{e^{- t}}{\sqrt{1 + t^{2}}} d t .$

Question Number 36433 Answers: 2 Comments: 1

valculate f(x)= ∫_0 ^2 (((x+2)^2 )/(√(x^2 +4x+5)))dx

$v a l c u l a t e f (x) = \int_{0}^{2} \frac{{(x + 2)}^{2}}{\sqrt{x^{2} + 4 x + 5}} d x$

Question Number 36432 Answers: 1 Comments: 1

calculate I = ∫_(−∞) ^(+∞) ((x+1)/((x^2 +1)^2 ))dx .

$c a l c u l a t e I = \int_{- \infty}^{+ \infty} \frac{x + 1}{{(x^{2} + 1)}^{2}} d x .$

Question Number 36431 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((x+1)/((x^2 +1)^2 ))dx

$c a l c u l a t e \int_{0}^{1} \frac{x + 1}{{(x^{2} + 1)}^{2}} d x$

Question Number 36430 Answers: 1 Comments: 1

find ∫ ((ln(x+x^2 ))/x^2 )dx

$f i n d \int \frac{l n (x + x^{2})}{x^{2}} d x$

Question Number 36429 Answers: 1 Comments: 1

let ϕ(λ) = ∫_(λ/π) ^(π/λ) (1+(1/x^2 ))arctan(x)dx with λ>0 1) find a simple form of ϕ(λ) 2) calculate ϕ^′ (λ).

$l e t φ (λ) = \int_{\frac{λ}{π}}^{\frac{π}{λ}} (1 + \frac{1}{x^{2}}) a r c t a n (x) d x w i t h λ > 0$ $1) f i n d a s i m p l e f o r m o f φ (λ)$ $2) c a l c u l a t e φ^{^{'}} (λ) .$

Question Number 36428 Answers: 2 Comments: 1

find ∫ (dx/(cos^4 x +sin^4 x))

$f i n d \int \frac{d x}{{c o s}^{4} x + {s i n}^{4} x}$

Question Number 36427 Answers: 1 Comments: 1

calculate ∫_(π/8) ^(π/6) (dx/(sin(2x)))

$c a l c u l a t e \int_{\frac{π}{8}}^{\frac{π}{6}} \frac{d x}{s i n (2 x)}$

Question Number 36426 Answers: 1 Comments: 0

find ∫ x^2 (√(1+x^3 )) dx

$f i n d \int x^{2} \sqrt{1 + x^{3}} d x$

Question Number 36425 Answers: 2 Comments: 2

calculate ∫_1 ^4 ((x(√x))/(x^2 −5x +4))dx

$c a l c u l a t e \int_{1}^{4} \frac{x \sqrt{x}}{x^{2} - 5 x + 4} d x$

Question Number 36424 Answers: 0 Comments: 2

find ∫ x^2 (√(x^2 −1))dx

$f i n d \int x^{2} \sqrt{x^{2} - 1} d x$

Question Number 36423 Answers: 0 Comments: 0

find ∫ (dt/(t(√(t^2 +t+1))))

$f i n d \int \frac{d t}{t \sqrt{t^{2} + t + 1}}$

Question Number 36422 Answers: 0 Comments: 1

find f(x)= ∫_0 ^x (t^2 +1)arctan(t)dt .

$f i n d f (x) = \int_{0}^{x} (t^{2} + 1) a r c t a n (t) d t .$

Question Number 36421 Answers: 0 Comments: 1

find ∫ (dx/(1+2(√(1−x))))

$f i n d \int \frac{d x}{1 + 2 \sqrt{1 - x}}$

Question Number 36420 Answers: 0 Comments: 0

let f(x)=∫_0 ^∞ (( arctan(xt^2 ))/(1+t^4 ))dt 1) calculate f^′ (x) 2) find a simple form of f(x).

$l e t f (x) = \int_{0}^{\infty} \frac{a r c t a n ({x t}^{2})}{1 + t^{4}} d t$ $1) c a l c u l a t e f^{^{'}} (x)$ $2) f i n d a s i m p l e f o r m o f f (x) .$

Question Number 36419 Answers: 0 Comments: 3

calculate ∫_0 ^∞ ((x^2 −1)/((2x^2 +3)^3 ))dx

$c a l c u l a t e \int_{0}^{\infty} \frac{x^{2} - 1}{{(2 x^{2} + 3)}^{3}} d x$

Question Number 36418 Answers: 1 Comments: 0

calculate ∫_(−3) ^4 ∣x^2 −2x−3∣dx

$c a l c u l a t e \int_{- 3}^{4} ∣ x^{2} - 2 x - 3 ∣ d x$

Question Number 36417 Answers: 1 Comments: 1

calculate ∫_2 ^6 (dx/((√(x+1)) +(√(x−1))))

$c a l c u l a t e \int_{2}^{6} \frac{d x}{\sqrt{x + 1} + \sqrt{x - 1}}$

Question Number 36416 Answers: 0 Comments: 0

calculate I = ∫_1 ^2 ((2x^3 +5x^2 −4x−7)/((x+2)^2 ))dx

$c a l c u l a t e I = \int_{1}^{2} \frac{2 x^{3} + 5 x^{2} - 4 x - 7}{{(x + 2)}^{2}} d x$

Question Number 36415 Answers: 0 Comments: 1

calculate I = ∫_0 ^(π/2) (x^3 +x)cos^2 xdx and J = ∫_0 ^(π/2) (x^3 +x)sin^2 xdx cslculate I and J .

$c a l c u l a t e I = \int_{0}^{\frac{π}{2}} (x^{3} + x) {c o s}^{2} x d x a n d$ $J = \int_{0}^{\frac{π}{2}} (x^{3} + x) {s i n}^{2} x d x$ $c s l c u l a t e I a n d J .$

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