4
votes
0answers
59 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
1
vote
4answers
73 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
2
votes
1answer
27 views

Problem understanding integral evaluation

I am having trouble understanding the evaluation of an integral. Do we just separate the integrals and evaluate them? Is it like normal integration? I have provided an example below taken from one of ...
0
votes
2answers
76 views

How to find the integral of $\dfrac{1-x}{(1+x)^2}$?

Can you help me with the $$ \int\frac{1-x}{(1+x)^2}dx$$ I was trying to integrate by parts but it doesn't work...
0
votes
1answer
60 views

How can I solve this indefinite integral?

Can someone please show me with steps on how to evaluate this indefinite integral?
2
votes
1answer
22 views

Trigonometric Substitution

Question: Use the substitution x=3sin(t) to evaluate the integral of I started by making a right triangle and solving for sin(t) and cos(t). sin(t)=x/3 and cos(t)=(sqrt(9-x^2))/3 Then, I solved ...
4
votes
1answer
44 views

Integration by Trig Substution - completely stuck

I'm trying to solve this integral, but after more than an hour I can't figure it out. I've outlined my thinking below. $$ \int \dfrac{dx}{x^2\sqrt{4x^2+9}} $$ If we let $\ a=3 $ and $\ b=2 $, the ...
1
vote
3answers
98 views

integral computation $\int_{-\infty}^{\infty} \frac{1}{(1+x+x^2)^2} dx $

Compute the following integral: $\int_{-\infty}^{\infty} \frac{1}{(1+x+x^2)^2} dx$. Can some one give me some hints on how to do this? I tried writing $(1+x+x^2)=f(x)$ and then multiplying and ...
0
votes
1answer
85 views

Trig Substitution Indefinite Integral help

I'm having trouble with this trig substitution problem. Please help me out. $$\int{\frac{ \sqrt{9x^2-289}}{x}}\;dx$$
5
votes
1answer
125 views

How to resolve this integration $\int\frac{dx}{1+x^2+\sin^2x}$?

I have tried Trigonometric Substitution, but I can´t get an already known function to be easy for integrate: $$\int\frac{dx}{1+x^2+\sin^2x}$$ I entered this on Wolfram and it gave me the same ...
2
votes
3answers
116 views

solving $\int x^7\sqrt{3+2x^4}dx$

I'm trying to solve $\int x^7\sqrt{3+2x^4}dx$ All I have so far is Let $u$ = $3+2x^4$ $du$ = $8x^3$ $dx$ $\frac{du}{8x^3}$ = $dx$ Therefore, $\int x^7\sqrt{u}$ $\frac{du}{8x^3}$ ...
0
votes
1answer
87 views

evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$

I'm trying to evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$ This is what I got so far: Let $u$ = $1-2x$ $x$ = $\frac{u-1}{-2}$ $du$ = $-2$ $dx$ $\frac{-du}{2}$ = $dx$ Therefore, ...
1
vote
1answer
39 views

Evaluating integral by parts.

Evaluate the following integral. $$ \int e^{2x}\sin{5x}\ dx $$ What I have tried : $ g(x) = \sin5x , f^{'}(x) = e^{2x} , f(x) = e^{2x} $ $$ \int e^{2x}\sin{5x}\ dx = e^{2x}\sin{5x} -\int e^{2x}\ ...
1
vote
1answer
36 views

Evaluating an integral by parts.

Evaluate the following integral. $$ \int x^2 e^x\ dx $$ What i have tried : $ f^{'}(x) = e^x , f(x) = e^x , g(x) = x^2$ $$ \int x^2 e^x\ dx = e^x\ x^2 - \int e^x\ 2x\ dx $$ $ f^{'}(x) = e^x , ...
0
votes
2answers
80 views

Integration - U substitution

I am given the following indefinite integral: $\int x\sqrt{1+2x}~dx$ using the substitution $u=\sqrt{1+2x}$ I am not sure where to begin, I know that $du=\dfrac{1}{\sqrt{1+2x}}$ But how can I get ...
0
votes
3answers
78 views

How do I determine the values of $r$ so $e^{t^r}$ has an antiderivative?

It's fairly easy to show for individual values of $r$, just plug in the value and try it, but is there any way to find, in general, which values of $r$ work so that $\int{e^{t^r}dt}$ can be expressed ...
1
vote
1answer
145 views

How can I solve$\int \sin^3(x)dx$?

I have to find the integral $$\int \sin^3xdx\\= \int \sin^2x \sin xdx\\= \int (1-\cos^2x) \sin xdx$$ Substitution: $$z=\cos x$$ $$\frac{dz}{dx} = -\sin x$$ $$-dz = \sin x dx$$ Now the above ...
1
vote
2answers
58 views

Evaluating the indefinite integral

I am having trouble understanding this homework question: $$\int \frac {dx}{cx+h} $$ So, what I thought I should do is... $$\int \frac {dx}{cx+h} $$ let $u$ be: $cx+h$ let $du$ be: $ 1\,dx $ ...
1
vote
5answers
132 views

$\displaystyle \int \frac{\sqrt{16-x^2}}{x} \mathrm{d}x$

The problem is $\displaystyle\int\frac{\sqrt{16-x^2}}{x}\mathrm{d}x$. I've attempted to use a trig substitution with $x=4\sin\theta$ and $\mathrm{d}x=4\cos\theta\ \mathrm{d}\theta$. This yields $ ...
-2
votes
2answers
61 views

Integrate using substitution method. [closed]

I need help integrating the following function: $$\int\frac{x^3+3x^2+2x-7}{2x+1}dx.$$
0
votes
0answers
59 views

Integral of $\int\cot (x)\cdot e^xdx$

I would like to integrate the following: $$\int\cot (x)\cdot e^xdx$$ What I tried to do is: set $\cot(x)$ as $u$ then $du$ is $\ln|\sin(x)|$ , about $e^x$ he stay as is. there is another way to do ...
4
votes
5answers
243 views

Integral of $\frac{1}{x^4+1}$ [duplicate]

Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated.
0
votes
2answers
265 views

A question on integration

I want to compute the following integral: $$\raise 1ex{\Large\int} \frac{\sqrt{\ln(x+\sqrt{1+x^2}})}{1+x^2}\,dx$$
0
votes
4answers
125 views

Evaluate the integral: $\displaystyle \int x \tan^{-1}\ x \,\mathrm{d}x$

Evaluate the integral: $$\int x\tan^{-1}x\,\mathrm{d}x$$ What I have so far: $$u = \tan^{-1}x$$ $$\mathrm{d}u = \frac{1}{1+x^2}\,\mathrm{d}x$$ $$\mathrm{d}v = x\,\mathrm{d}x$$ $$v = \frac{x^2}2$$ ...
0
votes
2answers
123 views

I need help finding integral of $\frac{1}{x^2+x^4}$

I don't know how can I solve this integral rational function. $$\int\frac{1}{x^2+x^4}\mathrm{d}x$$
0
votes
1answer
50 views

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.
2
votes
4answers
311 views

How to integrate $\int e^{-x}\arctan(e^x) \, dx$

After trying this multiple ways, I give up. Here's the integral: $$\int e^{-x}\arctan(e^x)\,dx$$ I have set $u=\arctan(e^x)$ and $dv=e^{-x}d\,x$ and have obtained $du=\dfrac{e^x \, dx}{1+e^{2x}}$ ...
2
votes
3answers
68 views

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
0
votes
2answers
80 views

Integration of $c(y^2)(1-y)^4$

Could anyone please help with integrating $f(y)=cy^2(1-y)^4$? Where $c$ is a constant.
0
votes
1answer
46 views

Primitive of $(\frac{1}{\sqrt2}v_0e^{-kt})\sqrt{1+e^{-kt}} $

I have a watertank with a waterflow given by: $$v(t) = (\frac{1}{\sqrt2}v_0e^{-kt})\sqrt{1+e^{-kt}} $$ as a function of $t$ where $v_0$ and $k$ are positive constants. I'm trying to define the ...
1
vote
2answers
119 views

Use Substitution or partial integration to solve $\int x\sqrt{1-x^2}\,dx$

I'm struggling with the following homework: Use Substitution or partial integration to solve $$\int x\sqrt{1-x^2}\,dx$$ Ok, so since we have a multiplication, partial integration seems like the ...
1
vote
1answer
5k views

Integral of sqrt{1-x^2} using Integration by parts

I was asked to solve this indefinite integral using Integration by parts. $$\int \sqrt{1-x^2} dx$$ I know how to solve if use the substitution $x=sin(t)$ but I'm looking for the Integration by parts ...
0
votes
2answers
134 views

Difficult Integral Question

I'm trying to evaluate the following integral; $$\int e^{(x^2 - z^2)} (2x \cos(2xz) - 2z \sin(2xz)) dz$$ I've tried splitting it up, and using integration by parts, but it just isn't coming out in a ...
-1
votes
2answers
179 views

An arbitrary collection of integration exercises [closed]

1) I got the $A=2$ and $B=-1$. I think I'm just having trouble with the integration part now. Evaluate the integral $$\int_0^1 \frac{x-6}{x^2-6x+8}\,dx.$$ 2) For this one I got that $A=7$, ...
2
votes
1answer
658 views

integral of $\int\frac{2\sin(2x)-\cos(x)}{6-\cos^2x -4\sin x}\mathrm{d}x$

so i got this problem $$\int\dfrac{2\sin 2x-\cos x}{6-\cos^2x -4\sin x}\mathrm{d}x$$ now this is what i tried $=\int\dfrac{4\sin(x)\cos(x)-\cos(x)}{6-(1-\sin^2x) -4\sin ...
6
votes
3answers
692 views

$\int\sqrt{1-\cos2x}~dx=$ ?

So here is the problem I'm working with $$\int\sqrt{1-\cos2x}~dx$$ I'm assuming that I'll need to use the trig identity $2\sin^2x=1-\cos2x$ . But where do I go from there? ...
1
vote
4answers
114 views

Integration of $\int_{}^{}\frac{1}{x(1+x)^3}dx $

we got this integration problem $$\int_{}^{}\frac{1}{x(1+x)^3}dx $$ it seems a fairly simple problem but what i am struggling with it is doing its partital fractions $\int_{}^{}\frac{1}{x(1+x)^3}dx ...
1
vote
2answers
155 views

Evaluating $\int{e^{x^{1/3}}dx}$

How can I get $$\int{e^{x^{1/3}}dx}$$ I think integrating by parts may work, but I can't figure out the exact way.
2
votes
2answers
79 views

integration of $\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x}$

$$\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x}$$ We got this integral problem here, now fractional powers are causing trouble in substitution. So I tried substituting for both $\sin^{1/2}x$ and ...
1
vote
2answers
72 views

Evaluating $\int\frac{1}{1+3\cos^2 x}dx$

I got this fairly simple looking problem $$\int\frac{1}{1+3\cos^2 x}dx$$ Although it looks simple enough but I am stuck on how to begin a hint might be enough to solve this problem. What I tried was ...
4
votes
2answers
132 views

Evaluating $\int x^x\ln(xe) \,\mathrm dx $

$$\int x^x\ln(xe) \,\mathrm dx $$ we got this problem so i seperated this as $\int x^x \ln(x) \,\mathrm dx $ + $\int x^x \ln(e) \,\mathrm dx $ it becomes $\int x^x \ln(x) \,\mathrm dx$ + $\int x^x ...
3
votes
4answers
108 views

Compute $\int x^2 \cos \frac{x}{2} \mathrm{d}x$

I am trying to compute the following integral: $$\int x^2 \cos \frac{x}{2} \mathrm{d}x$$ I know this requires integration by parts multiple times but I am having trouble figuring out what to do once ...
4
votes
4answers
212 views

Calculate $\int \frac{dx}{x\sqrt{x^2-1}}$

I am trying to solve the following integral $$\int \frac{dx}{x\sqrt{x^2-1}}$$ I did the following steps by letting $u = \sqrt{x^2-1}$ so $\text{d}u = \dfrac{x}{\sqrt{{x}^{2}-1}}$ then \begin{align} ...
1
vote
3answers
89 views

evaluation the integral

I am struggling to find the values of these integrals after trying many substitution it did't worked for me 1) $$ \int_{0}^{a}\frac{dx}{\sqrt{ax-x^2}} $$ 2) $$ \int_{}^{}\frac{3x^5dx}{1+x^{12}} $$
1
vote
3answers
91 views

Indefinite integral of $(2x+9)e^x$

What is the indefinite integral $\displaystyle\int (2x+9)e^x\,\mathrm dx$? Attempt: Integration by parts seems obvious. $u = 2x + 9, \mathrm du = 2$ $\mathrm dv = e^x, v = e^x$ $uv - \int ...
1
vote
2answers
72 views

Can you explain me this antiderivative?

Find the antiderivative of $\displaystyle \frac{e^{\frac{x}{2}}}{e^x+2e^{\frac{x}{2}}+5}$. The book suggests a switch of variables. Let $t=e^{\frac{x}{2}}$. And so $x=2\ln(t)$. The antiderivative ...
15
votes
1answer
608 views

How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$

How can I calculate $$ \int{\sec\left(x\right)\tan\left(x\right) \over 3x + 5}\,{\rm d}x $$ My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$ Now Using Integration ...
2
votes
3answers
148 views

Evaluate $\int \frac{1+\cos(x)}{\sin^2(x)}\,\operatorname d\!x$

I`m trying to solve this integral and I did the following steps to solve it but don't know how to continue. $$\int \dfrac{1+\cos(x)}{\sin^2(x)}\,\operatorname d\!x$$ $$\begin{align}\int ...
6
votes
1answer
69 views

Finding the antiderivative of $\frac 1{(1-x^m)^n}$,with $n,m\in\Bbb N$

For $m,n \in \Bbb N$, find the antiderivative of $g:(0,1)\rightarrow\mathbb{R}$ defined by: $$g(x)=\frac{1}{(1-x^m)^n}$$ Mathematica gives a result with functions we didn't learn about yet. The ...
3
votes
1answer
48 views

How do I proceed with this integral?

$$ \int (2x)\ cos(5x)\ dx$$ I put $u = 2x$ $du = 2\ dx$ $v = \frac{1}{5}sin(5x)$ $dv = cos(5x)\ dx$ Then I try $ uv - \int vdu $ $$ 2x \times \frac{1}{5}sin(5x) - \int 2\times\frac{1}{5}sin(5x)\ ...