Title: How can I integrate x^2 dx/ sqrt(x^2+21) without using the sqrt (a^2+u^2) form formulas? Post by: smither777 on Sep 15, 2012 How do I integrate this equation? I've tried utilizing tangent substitution but I can't seem to get the correct answer. Now, I know that there is a set formula for this but how would I do this if I didn't have said formula on an exam? Thanks.
Title: How can I integrate x^2 dx/ sqrt(x^2+21) without using the sqrt (a^2+u^2) form formulas? Post by: _biology on Sep 15, 2012 ? x^2 / ?(x^2+21) dx = by parts
? x d (?(x^2+21)) = x?(x^2+21) - ? ?(x^2+21) dx = x?(x^2+21) - 21? ?((x/?21)^2+1) d(x / ?21) = I continue for ? ?((x/?21)^2+1) d(x / ?21) using x / ?21 = sinh ( u) 1/?21 dx = cosh(u) du ? ?(sinh^2(u) +1) cosh(u) du = ? cosh^2 (u) du = (1/2)? (1+ cosh(2u)) du = (1/2)u + (1/4)sinh(2u) + C back to x and you get the answer Title: How can I integrate x^2 dx/ sqrt(x^2+21) without using the sqrt (a^2+u^2) form formulas? Post by: miggybling on Sep 15, 2012 x = sqrt(21) * tan(t)
dx = sqrt(21) * sec(t)^2 * dt x^2 * dx / sqrt(21 + x^2) => 21 * tan(t)^2 * sqrt(21) * sec(t)^2 * dt / sqrt(21 + 21tan(t)^2) => 21 * tan(t)^2 * sqrt(21) * sec(t)^2 * dt / (sqrt(21) * sqrt(1 + tan(t)^2)) => 21 * tan(t)^2 * sec(t)^2 * dt / sqrt(sec(t)^2) => 21 * tan(t)^2 * sec(t)^2 * dt / sec(t) => 21 * tan(t)^2 * sec(t) * dt To solve this, we'll need to integrate by parts: u = tan(t) du = sec(t)^2 * dt dv = sec(t) * tan(t) * dt v = sec(t) int(u * dv) => u * v - int(v * du) => tan(t) * sec(t) - int(sec(t)^3 * dt) => tan(t) * sec(t) - int(sec(t) * (1 + tan(t)^2) * dt) => tan(t) * sec(t) - int(sec(t) * dt) - int(sec(t) * tan(t)^2 * dt) int(tan(t)^2 * sec(t) * dt) = tan(t) * sec(t) - int(sec(t) * dt) - int(sec(t) * tan(t)^2 * dt) 2 * int(tan(t)^2 * sec(t) * dt) = tan(t) * sec(t) - int(sec(t) * dt) int(tan(t)^2 * sec(t) * dt) = (1/2) * (tan(t) * sec(t) - int(sec(t) * dt)) int(tan(t)^2 * sec(t) * dt) = (1/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C 21 * tan(t)^2 * sec(t) * dt => 21 * (1/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C => (21/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C x = sqrt(21) * tan(t) x / sqrt(21) = tan(t) x^2 / 21 = tan(t)^2 x^2 / 21 = sec(t)^2 - 1 (21 + x^2) / 21 = sec(t)^2 sec(t) = sqrt((21 + x^2) / 21) (21/2) * (tan(t) * sec(t) - ln|sec(t) + tan(t)|) + C (21/2) * ((x/sqrt(21)) * sqrt((21 + x^2) / 21) - ln|sqrt(21 + x^2) / sqrt(21) + x / sqrt(21)|) + C (21/2) * (x * sqrt(21 + x^2) / 21 - ln|x + sqrt(21 + x^2)| - ln(sqrt(21)) + C We can bring out ln(sqrt(21)) and multiply it with 21/2. This will give us some real constant value which we can add to the constant of integration C to create some new constant C (21/2) * ((1/21) * x * sqrt(21 + x^2) - ln|x + sqrt(21 + x^2)|) + C (1/2) * x * sqrt(21 + x^2) - (21/2) * ln|x + sqrt(21 + x^2)| + C Title: How can I integrate x^2 dx/ sqrt(x^2+21) without using the sqrt (a^2+u^2) form formulas? Post by: budi on Sep 15, 2012 x^2 dx/ sqrt(x^2+21)
(x^2+21 -21)/sqrt(x^2+21) = sqrt(x^2+21)- 21/sqrt(x^2+21).........Now use bothe formulas |