Calculate ∫2^7 4x^2 dx, given the following. ∫2^6 x^2 dx =...

Calculate \int_2^7 4x^2 dx, given the following. \int_2^6 x^2 dx = \frac{208}{3} \quad \int_6^7 x^2 dx = \frac{127}{3} \quad \int_2^6 x dx = 16 \int_2^7 4x^2 dx = ? (Type an integer or a simplified fraction.)

Transcript

00:01 The question is to evaluate integral 32x squared divided by x minus 21 times x plus 7 the whole square dx using partial fraction decomposition.

00:13 First consider the integrant 32x squared divided by x minus 21 times x plus 7 the whole square.

00:22 Using partial fraction decomposition, this can be split into the form a divided by x minus 21 plus b divided by x plus 7 plus c divided by x plus 7 the whole square.

00:38 Now the right hand side can be simplified using least common denominator as x plus 7 the whole square a plus x minus 21 times x plus 7 b plus x minus 21 c the whole divided by x minus 21 times x plus 7 the whole square from which we can simplify to get the equation 32 x square is equal to x square plus 14 x plus 49 times a plus x square minus 14 x minus 14 b plus x minus 147 b plus x minus 21c now equating the coefficients of x square we get the equation a plus b is equal to 32 equating the coefficients of x we get 14a minus 14 b plus c is equal to zero equating the constants we get 49 a minus 147 b minus 21 c is equal to zero now these three equations has to be solved simultaneously from the first equation we get a is equal to 32 minus b substituting the value of a in the second equation we get negative 28 b plus c is equal to negative 440 substituting the value of a in the last equation we get negative 196 b minus 21 c is equal to negative thousand five hundred sixty eight now multiplying this equation throughout by 21 we get negative 588 b plus 21 c is equal to negative 9 ,4088 now adding this equation and this equation together we get negative 58b plus 21 c is equal to negative 900 9 ,408 and the second equation is negative 196 b minus 21c is equal to negative thousand five hundred sixty eight and adding these two together we get negative 78 b is equal to negative 784 b is equal to negative 10 ,000 9776 from which we can solve for b to get the value 14.

03:30 Now substituting the value of b here, we get the value of a is equal to 18...

Question

Calculate \int_2^7 4x^2 dx, given the following. \int_2^6 x^2 dx = \frac{208}{3} \quad \int_6^7 x^2 dx = \frac{127}{3} \quad \int_2^6 x dx = 16 \int_2^7 4x^2 dx = ? (Type an integer or a simplified fraction.)

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