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I am trying to solve this integral: $$\int \cos\left(\ln\left(e^t+1\right)+at+b\right)dt $$ with $$a, b \in \Re$$ I can solve this: $$\int \cos\left(\ln\left(x\right)\right)dt $$ Defining: $u=\cos(\ln(x))$ and $v = \frac{1}{x}$. Then using the fomular two times: $$\int udv=uv-\int vdu$$ I can get the solution $$\frac {1}{2}x\cos(\ln(x))+ \frac {1}{2}x\sin(\ln(x)) +C$$

I also found the orther topic in our website The integral $\int\ln(x)\cos(1+(\ln(x))^2)\,dx$. But still have no idea with this.

I wonder is this non-integrable function ?!

user1101010
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Thiều Quang Minh Nhật
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  • Where this problem arise? For particular values of $a,b,t$ the integrand function is not even defined. Anyway any continuous function is integrable over closed intervals of $\mathbb{R}$, but I won't bet a penny on the fact that $(e^t+at+b)^i$ has a primitive which can be expressed in terms of elementary functions. – Jack D'Aurizio Dec 09 '18 at 06:40
  • I typed wrong equation. But i don't know how to rewrite my question. My question is about this: $\int cos\left(ln\left(e^t+1\right)+at+b\right)dt$ – Thiều Quang Minh Nhật Dec 09 '18 at 06:41
  • I derived this integration from the dynamic and kinematic equation of autonomous underwater vehicle (AUV) when I consider a turning motion of AUV. This is X coordinate. – Thiều Quang Minh Nhật Dec 09 '18 at 06:45
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    I tried both wxmaxima and sympy. Both cannot integrate it symbolically (of course, its antiderivative still exists since it is continuous). – Alex Vong Dec 09 '18 at 08:44
  • Does anyone have an idea? – Thiều Quang Minh Nhật Dec 13 '18 at 07:02

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