1
vote
3answers
63 views

Steps to solve $\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$?

What are the steps required to solve the following? $\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$ I'm not looking for anyone to do my homework. I usually have no problem figuring these things out -- ...
3
votes
5answers
108 views

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: ...
0
votes
1answer
43 views

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 ...
5
votes
2answers
155 views

How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?

How to integrate $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx$$ I tried the following approach: $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \int \frac{1}{\sin^4x + (1-\sin^2x)^2} \,dx = \int ...
4
votes
4answers
115 views

How to integrate $\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$?

How to integrate $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$$ ? I have: $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx = \int \frac{\cos x}{\sqrt{2\sin x\cos x}} \,dx = \frac{1}{\sqrt2}\int \frac{\cos ...
5
votes
2answers
184 views

How to solve $\int \frac{\,dx}{(x^3 + x + 1)^3}$?

How to solve $$\int \frac{\,dx}{(x^3 + x + 1)^3}$$ ? Wolfram Alpha gives me something I am not familiar with. I thought that the idea was using partial fractions because $x^3$ and $x$ are ...
2
votes
2answers
48 views

How to solve this first order non-linear ODE

I am struggling to solve this first order ODE. $$ u'\,^2 = a u^2 + b u^3$$ Mathematica gives me, $$ u(v) = \dfrac{a}{b} \mathrm{sech} \left( \dfrac{1}{2} \sqrt{a} \ (v + c) \right)^2 $$ So I ...
3
votes
3answers
112 views

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ? The numerator is a irreducible polynomial so I can't use partial fractions. I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int ...
4
votes
1answer
72 views

Solve $\int\frac{\sqrt{(x-5)(x+3)}}{(x-1)(x^2-25)}\ dx$

I have some problems with the task. How to evaluate $$\int\frac{\sqrt{(x-5)(x+3)}}{(x-1)(x^2-25)}\ \mathrm{d}x$$ I have absolutely no idea. Help me please. Thank you.
3
votes
2answers
59 views

Integral of $\sqrt{a^2+b^2t^2}$

I'm trying to calculate mass of some line and this is the integral needed to be solved. Wolfram shows me some way with the fuction sec and reduction methods and I don't know how to use these. is ...
0
votes
1answer
73 views

How to integrate $\int\frac{xe^{tan^{-1}x}}{(1+x^2)^\frac{3}{2}}\,dx$? [duplicate]

Original question: (updated question in section EDIT) How to evaluate the following integral? $$\int\frac{1}{(1+x^2)^\frac{3}{2}}\,dx$$ I tried substitution: $$ t = 1+x^2 = \varphi \\ ...
1
vote
2answers
54 views

a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
1
vote
2answers
92 views

Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
8
votes
1answer
158 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
1
vote
1answer
40 views

Ho To Perform U-Substitution On Given Intergal

$$\int{x^2\sqrt{2+x}}\ {dx}$$ I haven't been able to find what u should be in this integral, where should I start? I've gotten as far as: let $u = 2 + x$; $du=\frac{1}{x}dx$
5
votes
1answer
116 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
77 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 ...
1
vote
2answers
81 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
62 views

How can I solve this indefinite integral?

Can someone please show me with steps on how to evaluate this indefinite integral?
4
votes
1answer
47 views

Trigonometric Substitution

Question: Use the substitution $x=3\sin(t)$ to evaluate the integral of $\int\sqrt{9-x^2}\,\mathrm dx$. I started by making a right triangle and solving for $\sin(t)$ and $\cos(t)$. ...
4
votes
1answer
48 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
100 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
91 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
146 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
159 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
119 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
54 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
48 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
90 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
81 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
183 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
69 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
142 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
63 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
62 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
258 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
281 views

A question on integration

I want to compute the following integral: $$\raise 1ex{\Large\int} \frac{\sqrt{\ln(x+\sqrt{1+x^2}})}{1+x^2}\,dx$$
1
vote
3answers
151 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
150 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
416 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
77 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
82 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.
1
vote
2answers
126 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
6k 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
149 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
190 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
689 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
792 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? ...