# Tagged Questions

Question about finding the primitives of a given function, whether or not elementary.

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### Determine the indefinite integral: $I=\int \frac{\ln x}{x\sqrt{1-4\sin x-\ln^2x}}dx$

Determine the indefinite integral: $$I=\int\frac{\ln x}{x\sqrt{1-4\sin x-\ln^2x}}dx$$ Really I don't know how, hence please help me. I need a solution!
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### Express the indefinite integral $\int \sin(x^2)$ as a power series

What does this mean? I never saw this in my class/notes so i don't understand the conversion from integral to power series. Also if the integral were defined from $0$ to $1$, what new steps do I add? ...
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### Indefinite Integral Inverse Trigonometric Function

$$\int \frac{2}{9w^2+25}dw$$ I already know this will be equal to $\frac{1}{a} \arctan(x/a)$, but I don't know how to factor out the $9$. I only know how to take out the $2$.
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### How to integrate $\int{\frac{6x}{x^3+8}dx}$

I'm having some trouble solving this integral using partial fraction method: $$\int{\frac{6x}{x^3+8}dx}.$$ After expanding $x^3+8$ into $(x-2)(x^2+2x+4)$ and expanding the original integral into ...
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### Finding the integral of *3xsin(x/4) *

In the picture above is my working out for the question "Find the indefinite integral of ($3xsin(\frac{x}{4}))$" (Sorry I don't know how to use mathjax, i'l learn.) The bordered area is the ...
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### Help on indefinite integrals.

I can't figure out how to find the integral for $$\int e^{-3x}\sin(x)dx$$ I get to $$(e^{-3x})(-\cos(x))-e^{-3x}-\sin(x)$$ and I don't know what to do after. My teacher said the answer was ...
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### Help finding indefinite integral?

I need help finding the indefinite integral $$\int \ln \left(\frac{2}{x}\right)$$ I get $x\ln\left(\frac{2}{x}\right)+1$ but my teacher gave us $x\ln 2-x\ln x+x+c$ as the correct answer. I'm not sure ...
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### integrate the following equation (what am I doing wrong here 2)

Here is the equation: $$\int 3x \sqrt{1-2x^2}dt$$ Here is my answer: $$\dfrac14 \int (1-2x^2)^{1/2} . 3x = -\dfrac14 \dfrac{(1-2x^2)^{3/2}}{3/2} = -\dfrac14 \cdot \dfrac23 (1-2x^2)^{3/2} + c$$ ...
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### Problem with integrating $\int t^2\sqrt t\, dt$

Here is the equation: $$\int t^2 \sqrt{t}dt$$ Here is my answer: $$\int t^2 \sqrt{t} = \dfrac12 \int t^{1/2}2t = \dfrac12 \dfrac{t^{3/2}}{3/2} = \dfrac12 \cdot \dfrac23 t^{3/2} + c$$ whereas here is ...
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### Help with $\int \frac 1{\sqrt{a^2 - x^2}} \mathrm dx$

$\int \dfrac 1{\sqrt{a^2 - x^2}} \,dx$ In the second passage it become $\int \dfrac 1{a\sqrt{1 - \left(\frac xa\right)^2}}\,dx$ So can someone explain me what kind operation is done at ...
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### Indefinite integral with multiple values evaluated at endpoints to get definite integral

I am having trouble evaluating the definite integral $$\int_{-\pi}^{\pi} \frac{\sin\phi}{\pi\left( 1-\cos^2\Lambda\cos^2\phi \right)}d\Lambda$$ by the standard technique of the difference of value of ...
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### I need help finding integral of $\frac{1}{x^2+x^4}$

I don't know how can I solve this integral rational function. $$\int\frac{1}{x^2+x^4}\mathrm{d}x$$
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### Integration with substitution

I want to integrate (2x+1)/((x^2 - 6x + 14)^3) I'm guessing you use substitution but im unsure what to substitute, is it best to make u = x^2 or X^2-6x or even x^2 - 6x + 14 I find that it makes ...
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### Find the indefinite integral $\int\frac{(x+1)e^x}{x(1+xe^x)}dx$

Find the indefinite integral $$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$ I feel like this function does not have an anti-derivative in the form of elementary functions.
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### Find point between two intervals which equals the average value.

Find the point in the interval [5,9] at which the function $f(x)=17e^{3x}$ equals its average value on that interval. So I've got my function as follows ...
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### Average Value of a function with intervals [-1,2]

Find the Average Value of the function $f(x)=x^2+8x$ on the interval [-1,2]. So I've written it with the average value formula so I looks like this $f(x)=\frac{1}{3}(\frac{x^3}{3}-4x^2)$. Next I ...
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### Help me evaluate this integral

$\displaystyle \int \frac{\ln x}{x^2} \mathrm dx$ I just can't seem to figure this one out. I tried integrating by parts but I'm stuck.
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### Integration by trig substitution - Why can I draw a right triangle and use that to go back to the original variable?

Suppose we are given an integral of the form $\int f(x) dx$ with $f(x)$ defined on $[a;b]$. We can then use inverse substitution using a function $g(t)$ defined over an interval $[\alpha;\beta]$ ...
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### When we can change $\int$ and $\sum$ for indefinite integral?

I know, for example, that if the series $\displaystyle\sum_{n=1}^{\infty}f_n(x)$ consisting of integrable functions on a closed interval $[a, b] \subset \mathbb{R}$ converges uniformly on that closed ...
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### Indefinite intergral of $\int { \sqrt{ x^2-a^2} \over x } dx$

I need to integrate $$\int { \sqrt{x^2-a^2} \over x } dx$$ using substition, and show it equals $$\sqrt{x^2 - a^2} - a(\operatorname{arcsec} ({x\over a })) +c$$ I've tried $x=a\sin t$ but I ...
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### Evaluating $\int e^{-x}\tan(x) \ \mathrm dx$ .

I was told years ago by a visiting professor that this integral:$$\int e^{-x}\tan(x)dx$$ has an elementary form, but I have never been able to find it. Any suggestions? I don't think it's possible ...
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### how to solve this indefinite integral?

could anyone help me how to solve this indefinite integral? $\int{dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$ Thank You.
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### Find the indefinite integral:

I have to find the indefinite integral of (e^x)(cos x). I tried using integration by parts but I keep getting the same integral over and over again.
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### Can sign of integration by parts equation be changed to negative?

Integration by parts: $$\int u \:dv+\int v \: du=uv$$ Are there any circumstances under which the sign can be changed so that the following is true? $$\int u \:dv-\int v \: du=uv$$
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### Try to solve integral $\int \frac{3-7z}{21z^2-7}dz$

$$\int \frac{3-7z}{21z^2-7}dz$$ I would like to get some advice how to solve this integral
### Systematic method to evaluate $\int x(e^x) dx$
My calculus skills are pathetic. I haven't done the following integration exercise before, but this looks like it should easy to evaluate. $$\int x e^x\ \ \mathrm{dx}$$ I can probably ...
How can I solve the following indefinite integral containing inverse $\cosh$? Does it have any antiderivative? $$I(t,a) = \int \frac{\text{arccosh}\ t}{\sqrt{t^2-a^2}} dt.$$ The case $a = 1$ is ...