Given information:

The integral is ∫10(x−1)(x2+9)dx

Formula used:

  ∫1xdx=ln|x|+C

  ∫dxx2+a2=1atan−1(xa)

Calculation:

The partial function expansion of the integrand is,

  10(x−1)(x2+9)=Ax−1+Bx+Cx2+9

Here, A,B and C are constants. To find the value of A,B and C multiply both sides of the equation by (x−1)(x2+9)

  10=A(x2+9)+(Bx+C)(x−1)10=Ax2+9A+Bx2−Bx+Cx−C10=x2(A+B)+x(C−B)+9A−C

The coefficients of x must be equal and the constant terms are also equal. So

  A+B=0, C−B=0 and 9A−C=10

Solve this system of linear equation for A,B and C , by use of simple substitution

  0=A+B⇒A=−B0=C−B⇒C=B⇒C=−A10=9A−C⇒10=−A+A

So, A=1,B=−1 and C=−1

Each of the resultants partial function is easy to integrate. Evaluate integral on the right side of the equality. Use the substitution u=x−1 and v=x2+9 respectively. So,

  ∫10(x−1)(x2+9)dx=∫1x−1dx+∫−x−1x2+9dx

Integrate the function to get the final result.

  ∫10(x−1)(x2+9)dx=∫1x−1dx+∫−x−1x2+9dx=∫1x−1dx−∫x+1x2+9dx=∫1x−1dx−∫xx2+9dx−∫1x2+9dx=∫1udu−12∫1vdv−∫1x2+9dx=ln|u|−12ln|v|−13tan−1(x3)  [formula:∫dxx2+a2=1atan−1(xa)]=ln|x−1|−12ln|x2+9|−13tan−1(x3)+C

Therefore, the value of the given integral is ln|x−1|−12ln|x2+9|−13tan−1(x3)+C