Example Question - definite integrals
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Exploration of an Unusual Integral Expression
The image displays an integral that needs to be solved. The integral is written as: \[-2 \int_{0}^{t} 0.1t \, dt\] This integral is a bit unusual because the variable of integration, \( dt \), matches a variable within the integral's expression, \( 0.1t \). Normally, integrals like this one, with a variable as the upper limit of integration and the same variable in the integrand, would be approached using definite integrals. However, this appears to be an error since usually, the integrand should not contain the same variable as the limit of an integral without further context or special instructions like dealing with a parameter. Assuming that there is an error and that the variable in the integrand should be different from the variable of integration, let's proceed with the integral by replacing \( 0.1t \) with a constant \( k \) (since an integral with respect to \( dt \) should not have \( t \) as part of the integrand): \[-2 \int_{0}^{t} k \, dt\] Now let's integrate assuming that \( k \) is a constant: \[-2k \int_{0}^{t} dt = -2k[t]_{0}^{t} = -2k(t - 0) = -2kt\] If we revert back to using the original variable \( 0.1t \), but keep in mind it should be a constant with respect to \( t \), we would then shorthand write: \[-2kt\] But since this is against the general rules of calculus and may cause confusion, this solution is provided with caution and under the assumption that the integral's integrand was meant to represent a constant factor, and not the variable of integration \( t \).
Symmetry Property and King's Property in Definite Integrals
The given integral is: \[ \int_{0}^{\frac{\pi}{2}} \frac{\cos x}{(4 + \sin x)^2} \, dx \] A suitable method to solve this integral is to use a symmetry property known as the King's property in definite integrals, which uses the substitution \( x = \frac{\pi}{2} - t \) to exploit the symmetrical limits of integration from 0 to \( \frac{\pi}{2} \). Let's apply the King's property and see what we get: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos(\frac{\pi}{2} - t)}{(4 + \sin(\frac{\pi}{2} - t))^2} \, dt \] We know that \( \cos(\frac{\pi}{2} - t) = \sin t \) and \( \sin(\frac{\pi}{2} - t) = \cos t \), so substituting these in, we get: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin t}{(4 + \cos t)^2} \, dt \] If we now add the original integral and the transformed integral, we get: \[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\cos x}{(4 + \sin x)^2} + \frac{\sin x}{(4 + \cos x)^2} \right) dx \] Now, since the integrand is symmetric about the line \( y = x \), we can rewrite the integral in terms of a single variable: \[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\cos x + \sin x}{(4 + \sin x)^2} \, dx \] By observation or using trigonometric identities, notice that: \[ 4 + \sin x + \cos x = (4 + \sin x)(1 + \sec x) \] Now differentiate the denominator with respect to x. \[ d(4+\sin x) = \cos x \,dx \] Now, \[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{d(4 + \sin x)}{(4 + \sin x)^2} \] This can now be easily integrated using the formula for the integral of a derivative of a function over the function squared: \[ 2I = \left. -\frac{1}{4 + \sin x} \right|_{0}^{\frac{\pi}{2}} \] When we plug in the limits, we get: \[ 2I = \left( -\frac{1}{4 + \sin(\frac{\pi}{2})} \right) - \left( -\frac{1}{4 + \sin(0)} \right) \] \[ 2I = \left( -\frac{1}{5} \right) - \left( -\frac{1}{4} \right) \] \[ 2I = -\frac{1}{5} + \frac{1}{4} \] \[ 2I = \frac{1}{4} - \frac{1}{5} \] \[ 2I = \frac{5 - 4}{20} \] \[ 2I = \frac{1}{20} \] Hence, the value of the original integral is: \[ I = \frac{1}{2} \cdot \frac{1}{20} = \frac{1}{40} \]
Definite Integrals of Polynomial Functions
The image shows an integral which needs to be solved. The function to be integrated is a polynomial: 8x^3 - x^2 + 5x - 1. To solve the integral, we need to integrate each term individually with respect to x. Here's how you integrate each term: 1. ∫8x^3 dx = (8/4)x^4 = 2x^4 2. ∫(-x^2) dx = -(1/3)x^3 3. ∫5x dx = (5/2)x^2 4. ∫(-1) dx = -x Now, putting it all together: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C represents the constant of integration.
Evaluating Definite Integrals of Polynomial Functions using Power Rule
This is a definite integral of a polynomial function, which we can evaluate using the Power Rule for integration. The Power Rule states that the integral of x^n is (x^(n+1)) / (n+1) plus a constant (C), for any real number n ≠ -1. So let's evaluate your integral: ∫(8x^3 - x^2 + 5x - 1) dx Integration term by term: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) = 2 * x^4 ∫-x^2 dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) = - (1/3) * x^3 ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) = (5/2) * x^2 ∫-1 dx = -x Combining these results: 2x^4 - (1/3)x^3 + (5/2)x^2 - x Without limits of integration given, we cannot evaluate for specific numbers. If there were limits, you would substitute the upper limit into the result for x, then subtract the result from substituting the lower limit. Since no limits are provided, this is the indefinite integral result, and don't forget to add the constant of integration C: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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