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Integrals

Short Answer Type

281.

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

Long Answer Type

282.

Evaluate: $\int$ sin x sin 2x sin 3x dx

283.

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

284.

Prove that $\int_{0}^{\frac{π}{4}} \sqrt{\tan x} + \sqrt{\cot x} dx = \sqrt{2} . \frac{π}{2}$

285.
Evaluate $\int_{1}^{3} 2 x^{2} + 5 x dx$ as a limit of sum.

$\int_{1}^{3} 2 x^{2} + 5 x dx Here, a = 1, b = 3, f (x) = 2 x^{2} + 5 x ∴ n h = b - a = 3 - 1 = 2 Now \int_{a}^{b} f (x) dx = \lim_{h \to 0} h f (a) + f (a + h) + f (a + 2 h) + . . . . . + F (a + (n - 1) h) ∴ \int_{1}^{3} 2 x^{2} + 5 x dx = \lim_{h \to 0} h f (1) + f (1 + h) + f (1 + 2 h) + . . . . . + F (1 + (n - 1) h)$

$= \lim_{h \to 0} h [2 (1)^{2} + 5 (1) + 2 (1 + h)^{2} + 5 (1 + h) + (2 (1 + 2 h)^{2} + 5 (1 + 2 h)) . . . . . . . + (2 (1 + (n - 1) h)^{2} + 5 (1 + (n - 1) h))] = \lim_{h \to 0} h [7 + (2 h^{2} 9 h + 7) + (8 h^{2} + 18 h + 7) + . . . . . . . . . . + {(2 (n - 1)}^{2} h^{2} + 9 (n - 1) h + 7)] = \lim_{h \to 0} h [7 n + 2 h^{2} (1^{2} + 2^{2} + . . . . . . . . + (n - 1)^{2}) + 9 h (1 + 2 + . . . . . . . . + (n - 1)] = \lim_{h \to 0} h [7 n + 2 h^{2} \frac{n (n - 1) (2 n - 1)}{6} + 9 h \frac{n (n - 1)}{2}]$

$= \lim_{h \to 0} h [7 n h + 2 \frac{n h (n h - h) (2 n h - h)}{6} + 9 \frac{n h (n h - h)}{2}] = \lim_{h \to 0} h [14 + 2 \frac{2 (2 - h) (4 - h)}{6} + 9 \frac{2 (2 - h)}{2}] = 14 + \frac{16}{3} + 18 = \frac{112}{3}$

Multiple Choice Questions

286.

The integral $integral fraction numerator 2 straight x to the power of 12 space plus space 5 straight x to the power of 9 over denominator left parenthesis straight x to the power of 5 space plus space straight x cubed space plus space 1 right parenthesis cubed end fraction space dx$ is equal to

$fraction numerator negative straight x to the power of 5 over denominator left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C$
$fraction numerator straight x to the power of 10 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C$
$fraction numerator straight x to the power of 5 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C$
$fraction numerator straight x to the power of 5 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C$

141 Views

287.

Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (−1, −2), then which one of the following is a vertex of this rhombus?

(−3, −9)
(−3, −8)
(1/3, -8/3)
(1/3, -8/3)

536 Views

288.

The centres of those circles which touch the circle, x²+y²−8x−8y−4=0, externally and also touch the x-axis, lie on:

a circle
an ellipse which is not a circle
a hyperbola.
a hyperbola.

245 Views

289.

The integral $integral fraction numerator dx over denominator straight x squared space left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction space equals$

$open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C$
$left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C$
$negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C$
$negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C$

123 Views

290.

The integral $integral subscript 2 superscript 4 fraction numerator log space straight x squared over denominator log space straight x squared space plus space log space left parenthesis 36 minus 12 space straight x plus straight x squared right parenthesis end fraction dx$ is equal to