Re: Brad's questions at Yahoo Answers regarding finding anti-derivatives
Hello Brad,
1.) $$I=\int-8(4x-1)^{\frac{1}{2}}\,dx$$
Let's use the substitution:
$$u=4x-1\,\therefore\,du=4\,dx$$
And we may now state:
$$I=-2\int u^{\frac{1}{2}}\,du$$
Using the power rule for integration, we may write:
$$I=-2\left(\frac{u^{\frac{3}{2}}}{\frac{3}{2}} \right)+C=-\frac{4}{3}u^{\frac{3}{2}}+C$$
Back-substituting for $u$, there results:
$$I=-\frac{4}{3}(4x-1)^{\frac{3}{2}}+C$$
2.) $$I=\int -2(3x^4-5)^2\,dx$$
Bringing the constant factor in the integrand out front and expanding the binomial, we have:
$$I=-2\int 9x^8-30x^4+25\,dx$$
Applying the power rule term by term, we find:
$$I=-2\left(9\frac{x^9}{9}-30\frac{x^5}{5}+25x \right)+C$$
$$I=-2\left(x^9-6x^5+25x \right)+C$$