The value of int_0^7 4xdx is the same as the area of a ,vv0^(8). Finding the appropriate dimensions gives int_0^7 4xdx= The value of int_0^(16) 4xdx is the same as the area of a Select an answer 0 . Finding the appropriate dimensions gives int_0^(16) 4xdx= The value of int_7^(16) 4xdx is the same as the area of a Select an answer 0 . Using the previous results gives int_7^(16) 4xdx= The value of 4 d is the same as the area of a Finding the appropriate dimensions gives 4x dx The value of 4d is the same as the area of a Select an answer Finding the appropriate dimensions gives 4x dx The value of 4 d is the same as the area of a Select an answer Using the previous results gives 4x dx
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To find this area, we can calculate the definite integral: \int_0^7 4xdx = [2x^2]_0^7 = 2(7)^2 - 2(0)^2 = 2(49) = 98 Show more…
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EXAMPLE 5 Evaluate ∫ (10x² - 9x + 16) / (x³ + 4x) dx. SOLUTION Since x³ + 4x = x(x² + 4) can't be factored further, we write (10x² - 9x + 16) / (x(x² + 4)) = A / [ ] + (Bx + C) / [ ]. Multiplying by x(x² + 4), we have 10x² - 9x + 16 = A(x² + 4) + ([ ])x = [ ] + Cx + 4A. Equating coefficients, we obtain A + B = 10 C = -9 4A = 16. Thus A = [ ], B = [ ], and C = -9 and so ∫ (10x² - 9x + 16) / (x³ + 4x) dx = ∫ ([ ]/x + ([ ]x - 9) / (x² + 4)) dx. In order to integrate the second term we split it into two parts: ∫ ([ ]x - 9) / (x² + 4) dx = ∫ ([ ]x) / (x² + 4) dx - ∫ 9 / (x² + 4) dx. We make the substitution u = x² + 4 in the first of these integrals so that du = 2x dx. We evaluate the second integral by means of this formula with a = 2. (Remember to use absolute values where appropriate.) ∫ (10x² - 9x + 16) / (x(x² + 4)) dx = ∫ 4/x dx + ∫ ([ ]) dx - ∫ 9 / (x² + 4) dx = [ ] + K
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