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I found this example in a textbook: $$\int e^{\cos^2 x}\sin2x dx$$

There are also results, but I am not even close to that...

Fermat
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user984621
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5 Answers5

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Try using $\sin 2x= 2\sin x \cos x$ and $d(\cos x)=-\sin x \,dx$, then integrate by parts.

MPW
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dneug
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$$u= \cos^2(x)$$ Then $$I=-\int e^{u}du=-e^{u}+C=-e^{\cos^2(x)}+C$$

Fermat
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Since $\sin(2x) = 2 \sin(x) \cos(x)$ and \begin{align} \partial_{x} e^{\cos^{2}(x)} &= e^{\cos^{2}(x)} \cdot \partial_{x} ( \cos^{2}(x) ) \\ &= - 2 \sin(x) \cos(x) \cdot e^{\cos^{2}(x)} = - \sin(2x) e^{\cos^{2}(x)} \end{align} which leads to \begin{align} \int \sin(2x) \ e^{\cos^{2}(x)} \ dx = - e^{\cos^{2}(x)}. \end{align}

Leucippus
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  • Mighty peculiar notation for a simple derivative, isn't it? Why not $\frac{d}{dx}$ instead of the odd $\partial_x$? – MPW May 12 '14 at 19:49
  • in single variable calculus: $\partial_{x} = \frac{\partial}{\partial x} = \frac{d}{dx}$. – Leucippus May 12 '14 at 19:55
  • And ∂_x is less typing than d/dx. – Urgje May 12 '14 at 21:54
  • @Leucippus: I understand the notation, of course, but it seems inappropriate for this question. I doubt the OP knows it given the nature of the question. – MPW May 12 '14 at 23:57
  • @Urgje: Let's see... "\frac d{dx}" has 11 characters, "\partial_x" has 10 characters. Quite a savings, that. ;) – MPW May 13 '14 at 00:01
  • It is since I use scientific Word. – Urgje May 13 '14 at 10:45
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Even Better Hint: Try putting $u=\cos^2 x$ and consider $\int e^u\,du$

MPW
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Use the identity that $$\frac{d}{dx}e^{f(x)}=\frac{df(x)}{dx}e^{f(x)}\Rightarrow \int \frac{df(x)}{dx}e^{f(x)}dx$$

Setting $f(x)=\cos^2x$, can you find $\frac{df(x)}{dx}$ using the chain rule? How does it compare with the integral you have to evaluate?

Also, note that $\sin(2x)=2\sin(x)\cos(x)$ .

Alijah Ahmed
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