Advertisements
Advertisements
प्रश्न
Evaluate the following integral:
Advertisements
उत्तर
\[\int_{- 1}^1 \left| 2x + 1 \right| d x\]
\[We\ know\ that\, \left| 2x + 1 \right| = \begin{cases} - \left( 2x + 1 \right) &, &- 1 \leq x \leq - \frac{1}{2} \\\left( 2x + 1 \right) &, &- \frac{1}{2} < x \leq 1\end{cases}\]
\[ \therefore I = \int_{- 1}^\frac{- 1}{2} - \left( 2x + 1 \right) d x + \int_{- \frac{1}{2}}^1 \left( 2x + 1 \right) d x\]
\[ \Rightarrow I = - \left[ x^2 + x \right]_{- 1}^\frac{- 1}{2} + \left[ x^2 + x \right]_{- \frac{1}{2}}^1 \]
\[ \Rightarrow I = - \frac{1}{4} + \frac{1}{2} + 1 - 1 + 1 + 1 - \frac{1}{4} + \frac{1}{2}\]
\[ \Rightarrow I = \frac{5}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/4) (sin x + cos x)/(9+16sin 2x) dx`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
`int dx/(e^x + e^(-x))` is equal to ______.
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integrals as limit of sums:
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
