Question

11.Vx' -8x +7 dx is equal to

Answer 1

Let, the given integral be f( x ),

f( x )= ∫ x 2 −8x+7 dx

In order to make the expression, ( a x 2 +bx+c ) a perfect square, addition and subtraction of ( b 2a ) 2 is required.

f( x )= ∫ ( x 2 −2( 4 )x+7 ) dx = ∫ ( x 2 −8x )+7 dx

Add and subtract ( 8 2×1 ) 2 in the expression to make it a perfect square.

f( x )= ( x 2 −2.4x+ ( 4 ) 2 − ( 4 ) 2 )+7 dx = ∫ ( x−4 ) 2 −16+7 dx = ∫ ( x−4 ) 2 −9 dx = ∫ ( x−4 ) 2 − ( 3 ) 2 dx

Further, simplify the function by using the formula.

∫ x 2 − a 2 dx= 1 2 x x 2 − a 2 − a 2 2 log| x+ x 2 − a 2 |+C

Integrate the function by using the above formula.

f( x )= ∫ ( x−4 ) 2 − ( 3 ) 2 dx = ( x−4 ) 2 ( x−4 ) 2 − ( 3 ) 2 − ( 3 ) 2 2 log| x−4+ ( x−4 ) 2 − ( 3 ) 2 |+C = ( x−4 ) 2 x 2 −8x+7 − 9 2 log| x−4+ x 2 −8x+7 |+C

Thus, out of all the four options, option (D) is correct.