# Tagged Questions

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### What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
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### Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
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### Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
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### Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
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### Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
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### How to evaluate the following integral? $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
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### how to calculate integrate about Heaviside

everyone,here I have a question about how to calculate $$\int e^t H(t) dt$$ where $H(t)$ is Heaviside step function thank you for your answering!!
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### Indefinite integral of $x^x$

I've seen many many questions on the internet with answer that it cannot be done with elementary functions. Now I did this integration myself and got a pretty nice result. Since I've seen so many ...
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### Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
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### Evaluate $\int \sqrt{1-x^2}\,dx$

I have a question to calculate the indefinite integral: $$\int \sqrt{1-x^2} dx$$ using trigonometric substitution. Using the substitution $u=\sin x$ and $du =\cos x\,dx$, the integral becomes: ...
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### Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.
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### How to evaluate the following integral? $\int \ln(e^x + c)~\mathrm dx$

I can't seem to find an answer for this kind of integration, and I'd like to know if there is an answer for it, and if yes what is it. $$\int \ln(e^x + {c})~\mathrm dx\,,$$ where $c$ is a constant. ...
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### Integrating powers of linear and quadratic functions

How can I integrate function such as $(x+9)^3$? I obviously know that I can expand the function and integrate it normally. However, that is possible and feasible only as it is of third degree. What if ...
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### How to evaluate the following indefinite integral? $\int\frac{1}{x(x^2-1)}dx.$

I need the step by step solution of this integral please help me! I can't solve it! $$\int\frac{1}{x(x^2-1)}dx.$$
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### How do I solve this Integral using an infinite series

I'm supposed to use an infinite series to solve $$\int\frac{e^{2x}}{x}dx$$ How do I solve this? I know the answer already, I don't even have to use infinite series to solve this, however it was ...
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### Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1$

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1$$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
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### Difficult Integral in functional basis

Let $$g(x)=\int f\prime(x)\left[\frac{4}{3}x^2+4x^3+(2x^2+4x^3)f(x)+6x^2f^2(x)+xf^3(x)\right]dx$$ express $g(x)$ in terms of $\{1,x,x^2,x^3,....\}$ and $\{f(x),f^2(x),f^3(x),...\}$. Is there a clever ...
$$\int{ \frac{1}{(3t-1)(t+1)(t-2)}}{dt}$$ How many ways are there to solve this integral without using partial fractions? Thank you.