<p>\( \int f(x) \,dx = \int \frac{3x-5}{x^2+2x+1} \,dx \)</p> <p>Let's factor the denominator \( x^2 + 2x + 1 = (x+1)^2 \).</p> <p>Now, split the fraction into two parts: \( \frac{3x-5}{(x+1)^2} = \frac{Ax+B}{x+1} + \frac{C}{(x+1)^2} \)</p> <p>Find A, B, and C by multiplying both sides by \( (x+1)^2 \) and equating coefficients:</p> <p>\( 3x - 5 = Ax + B(x+1) + C \)</p> <p>Solve for A, B, and C, we get \( A = 3, B = -3, C = -2 \).</p> <p>Now integrate each term separately:</p> <p>\( \int \frac{3x-3}{x+1} \,dx - \int \frac{2}{(x+1)^2} \,dx \)</p> <p>For the first term, simplify to \( \int 3 - \frac{3}{x+1} \,dx \)</p> <p>Integrate directly:</p> <p>\( \int 3 \,dx - 3\int \frac{1}{x+1} \,dx - \int \frac{2}{(x+1)^2} \,dx \)</p> <p>The integrals are:</p> <p>\( 3x - 3\ln|x+1| - \frac{2}{x+1} + C \)</p> <p>Where \( C \) is the constant of integration.</p>
To solve this definite integral, we need to integrate each term separately within the bounds from -1 to 0. Let's integrate the function \( f(x) = 2x - e^x \). The integral of \( 2x \) is \( x^2 \), and the integral of \( -e^x \) is \( -e^x \), since the derivative of \( e^x \) with respect to \( x \) is \( e^x \). Let's integrate and then apply the bounds: \[ \int_{-1}^{0} (2x - e^x) dx = \left[ x^2 - e^x \right]_{-1}^{0} \] Now, applying the Fundamental Theorem of Calculus, we evaluate this antiderivative at the upper bound x=0 and subtract the value of the antiderivative at the lower bound x=-1: At x=0: \[ x^2 - e^x = 0^2 - e^0 = 0 - 1 = -1 \] At x=-1: \[ x^2 - e^x = (-1)^2 - e^{-1} = 1 - \frac{1}{e} \] Now subtract the value at x=-1 from the value at x=0: \[ (-1)-(1 - \frac{1}{e}) = -1 - 1 + \frac{1}{e} = -2 + \frac{1}{e} \] Hence the value of the definite integral is: \[ \int_{-1}^{0} (2x - e^x) dx = -2 + \frac{1}{e} \]
The question in the image is asking for the integral of the given function with respect to x: ∫(8x^3 - x^2 + 5x - 1) dx To solve this indefinite integral, you would integrate each term separately. The integral of a sum is the sum of the integrals, so you can integrate each term individually. The integral of 8x^3 with respect to x is (8/4)x^(3+1) = 2x^4. The integral of -x^2 with respect to x is -(1/3)x^(2+1) = -1/3 x^3. The integral of 5x with respect to x is (5/2)x^(1+1) = 5/2 x^2. The integral of -1 with respect to x is -x. Putting it all together, the indefinite integral of the function is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C is the constant of integration.
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