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I have a few homework problems, and they're asking me to find the antiderivative. They're quite tough, and I'm really confused. Can anyone be of some help on these few?
integral(4x-5/x^3)dx (/ = over.)
integral 3x over (x^2+4)dx. On this one, the 3x isn't in parentheses.
integral 5xsec^2(x^2)dx
integral (4x-5/x^2)dx
integral e^4x dx
integral sin(x)cos^3(x)dx
Any help on ANY of these is greatly appreciated. I'm really lost (Worried)
(1) $\displaystyle \frac{3x}{x^2+4}$
We know that $\displaystyle \frac{d}{dx}(\ln(x^2+4))= \frac{2x}{x^2+4}$, so all we need to do is find a fractional number to multiply/divide $\displaystyle \ln(x^2+4)$ with to get the antiderivative.
(2) $\displaystyle e^{4x}$
We know that $\displaystyle \frac{d}{dx}(e^{4x})=4e^{4x}$, so you simply need to multiply $\displaystyle e^{4x}$ with $\displaystyle \frac{1}{4}$ to get the antiderivative.
(3) $\displaystyle \sin(x)\cos^3(x)$
$\displaystyle \frac{d}{dx}(\cos^4(x)) = 4\cos^3(x)\cdot -\sin(x)$
Again, look for the appropriate number to multiply/divide with.
hi
$\displaystyle \int (4x-\frac{5}{x^{3}}) \, dx = 2x^{2} +\frac{5}{2}x^{-2} + C \; ,\, C\in \mathbb{R} $
$\displaystyle \int (\frac{3x}{x^{2}+4}) \, dx = \frac{3}{2}\cdot ln(x^{2}+4) + C $
Some of these are actually quite basic I am afraid.
