Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [indefinite-integrals]

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

5
votes
5answers
1k views

Finding $\int e^{2x} \sin{4x} \, dx$

Finding $$\int e^{2x} \sin 4x \, dx$$ I think I should be doing integration by parts... If I let $u=e^{2x} \Rightarrow du = 2e^{2x}$, $dv = \sin{4x} \Rightarrow v = -\frac{1}{4} \cos{4x}$ $\int{ e^{...
3
votes
4answers
221 views

Integration Problem: $\int \frac{x^2+1}{x^5-1}dx$

$$\int \frac{x^2+1}{x^5-1}dx$$ I am unable to integrate it, nothing works. Yes, I can use partial fraction but who remembers factorization of $x^5-1$, I need a better way of doing this.
14
votes
5answers
581 views

Evaluate $\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx$

I am unable to solve the following integral: $$\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx.$$ Tried putting $t=\tan{x}$ so that numerator has $\sec^{2}{x}$ but that doesn't help.
7
votes
4answers
5k views

Help finding integral: $\int \frac{dx}{x\sqrt{1 + x + x^2}}$

Could someone help me with finding this integral $$\int \frac{dx}{x\sqrt{1 + x + x^2}}$$ or give a hint on how to solve it. Thanks in advance
10
votes
2answers
18k views

Indefinite Integral for $\cos x/(1+x^2)$

I have been working on the indefinite integral of $\cos x/(1+x^2)$. $$ \int\frac{\cos x}{1+x^2}\;dx\text{ or } \int\frac{\sin x}{1+x^2}\;dx $$ are they unsolvable(impossible to solve) or is there a ...
4
votes
2answers
1k views

Evaluating $ \int \frac1{\sqrt{-x^{2} - 4x}}dx$

I am getting a sign error when evaluating: $$ \int \dfrac {1} {\sqrt{-x^{2} - 4x}}dx$$ I completed the square in the denominator leaving me: $$\int \dfrac {1} {\sqrt{-x^{2} - 4x + 4 - 4}}dx$$ $$\...
4
votes
4answers
1k views

Evaluating $\int \dfrac {2x} {x^{2} + 6x + 13}dx$

I am having trouble understanding the first step of evaluating $$\int \dfrac {2x} {x^{2} + 6x + 13}dx$$ When faced with integrals such as the one above, how do you know to manipulate the integral ...
2
votes
1answer
237 views

Equivalent to $\begin{align}\int \cos{\left(2x\right)} \ dx\end{align}$?

For some reason, I am lost on one part of this integration problem: $\begin{align}\int \cos{\left(2x\right)} \ dx\end{align}$ $u = 2x$ $du = 2 \ dx$ $\frac{1}{2} du = dx$ ...
2
votes
3answers
1k views

Evaluating $\int \frac{5-e^{x}}{e^{2x}}dx$ without Partial Fractions

I am trying to evaluate $$\int \frac{5-e^{x}}{e^{2x}} \mathrm dx$$ I tried rewriting the integral by throwing $e^{2x}$ up on top and using $$u=e^{x}$$ $$du = e^{x} dx$$ I then tried another ...
2
votes
5answers
2k views

Which formula do I use to integrate $ \int {\sqrt{x^2 + 81} \over 2} \,dx $

I am having trouble with a question really need help please. $$ \int {\sqrt{x^2 + 81} \over 2} \,dx $$ I thought about taking the square root off and turning the question into $\frac 12 \int (x^2 +...
5
votes
3answers
4k views

Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$

I am working on an integration by parts problem that, compared to the student solutions manual, my answer is pretty close. Could someone please point out where I went wrong? Find $\int e^{2\theta} \...
1
vote
1answer
72 views

Rewriting a function so it is in a form that can be easily integrated.

I am trying to find an integral of the function $f(x)=\frac{2}{3}x+3$. I am a bit stumped on how to find the integral of $\frac{2}{3}x$, specifically. I looked at the answer thinking I could work ...
2
votes
2answers
217 views

Methodology for Integration by Parts

I am looking at an example of integration-by-parts in my Calculus book, and there is one thing that I do not understand: Prove the reduction formula: $$\int \sin^n x \ dx = -\frac{1}{n} \cdot \cos x \...
1
vote
3answers
1k views

How can this indefinite integral be solved without partial fractions?

At first I had no doubt that I will have to use partial fractions on this integral: \begin{equation} \int \frac{3x^2+4}{x^5+x^3}dx \end{equation} I split it into two integrals and one of them give me ...
1
vote
2answers
174 views

Where to start integration of trigonometric function? [duplicate]

Possible Duplicate: Evaluating $\int P(\sin x, \cos x) \text{d}x$ I am stuck with this integral: \begin{equation} \int \frac{dx}{(3+\cos^2x) \cdot \tan x} \end{equation} I tried playing with ...
3
votes
3answers
316 views

It is possible to integrate this function?

I heard that that there are some unintegratable functions and I want to as if this one is not one of them? \begin{equation} \large \int \frac{t}{t+1}dt \end{equation} Got this by trying to solve ...
3
votes
2answers
2k views

Integrate using Partial Fraction decomposition, completing the square

The given problem is $\int{x\over x^3-1}dx$. I know this equals $${1\over3}\int {1\over x-1}-{x-1\over x^2+x+1}dx,$$ which can be separated into $${1\over3}\int {1\over x-1}dx - {1\over3}\int{x+(...
4
votes
3answers
232 views

Finding an indefinite integral

I have worked through and answered correctly the following question: $$\int x^2\left(8-x^3\right)^5dx=-\frac{1}{3}\int\left(8-x^3\right)^5\left(-3x^2\right)dx$$ $$=-\frac{1}{3}\times\frac{1}{6}\left(...
2
votes
1answer
315 views

Integration of rational functions..

Some rational function is giving me some trouble... \begin{aligned} \ \int \frac {x^2-9x+16}{(x-1)(x^2+6x-7)} dx \end{aligned} I simplified it like so: \begin{aligned} \ \int \frac {x^2-9x+16}{(x-1)...
4
votes
1answer
136 views

Integral two solutions

at school, we solved this integral and the solution we got was Wolfram Alpha gave a different solution Are these two solutions equal?
1
vote
2answers
7k views

Using double angle formulas in integration, trouble following an example.

I have just started looking at integration and I am having trouble understanding what has been done in one of the examples in the book I am working through. It involves using the double angle ...
-2
votes
2answers
177 views

How to deal with multilevel degree inside of an indefinite integral?

The integral in question looks like this: \begin{aligned} \large \ \int x^3 \cdot e^{8-7x^4} dx \end{aligned} I tried using u-substitution on all the part before dx and it eliminated e (with all its ...
3
votes
2answers
37k views

How to deal with multiplication inside of integral?

I have an undefined integral like this: \begin{aligned} \ \int x^3 \cdot \sin(4+9x^4)dx \end{aligned} I have to integrate it and I have no idea where to start. I have basic formulas for integrating ...
15
votes
3answers
8k views

What's the connection between the indefinite integral and the definite integral?

I want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...
26
votes
1answer
1k views

Proof that $\int x^x dx$ can't be done in terms of elementary functions?

Is there any easy proof that $\int x^x dx$ can't be done in terms of elementary functions? I know this is true, because the Risch algorithm gives a decision process for integration in terms of ...
16
votes
1answer
702 views

How do experts mentally classify indefinite integrals?

Integration is as much an art as a science. Someone who is an expert looks at an indefinite integral and classifies it in a different way than someone who is a beginner. E.g., the beginner may just ...
10
votes
3answers
7k views

What is the integral of $e^x \tan(x)$?

What is the integral of $e^x \tan(x)$? Using basic theorems it is difficult to get I think.
4
votes
1answer
291 views

How do you integrate $\cos(x^n)$, specifically for $n=-1$?

How does one integrate $\cos(x^{-1})$? I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put $\...
19
votes
5answers
99k views

Is there a rule of integration that corresponds to the quotient rule?

When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives ...
6
votes
3answers
4k views

Integrate $\int \sin (2x) \cos (2x)\,dx$

I have $$\int \sin(2x) \cos (2x)\,dx = \frac12 \int \sin(4x)\,dx = -\frac18 \cos(4x),$$ but I also have $$\int \sin(2x) \cos (2x)\,dx = \frac12 \int \sin 2x \cdot 2 \cos 2x \, dx = \frac14 \sin^2(...
13
votes
3answers
909 views

Evaluate $\int \sqrt{1+x^{\frac{3}{2}}} \operatorname d x$

I can't figure out how to integrate $$\int \sqrt{1+x^{\frac{3}{2}}} \operatorname d x$$ I've tried substitution by letting $u = x^3$, but it didn't go anywhere. I also tried to integrate using a ...
8
votes
4answers
2k views

Trig substitution integration of $\int1/(x^2\sqrt{x^2 - 9}) dx$ - stuck on a problem

I am getting stuck on this trig substitution problem. $$\int\frac1{x^2\sqrt{x^2 - 9}}~\mathrm dx.$$ $$x = 3 \sec\theta,\qquad\theta = \sec^{-1} \sqrt{\frac{x^2}{9}},\qquad\mathrm dx = \sec\theta\tan\...
1
vote
2answers
167 views

Integration - Primitives - Antiderivatives

Please help to calculate: $$\int\sqrt {{r}^{2}-{x}^{2}}{dx},\quad x\in[0,r]$$ Do any method of trigonometric substitution? Thanks.
7
votes
4answers
313 views
3
votes
2answers
346 views

How to calculate $\int \sqrt{(\cos{x})^2-a^2} \, dx$

How to calculate: $$\int \sqrt{(\cos{x})^2-a^2} \, dx$$
3
votes
3answers
190 views

Integral with a substitution

I must calculate a following integral $$\int \frac{dx}{x^{2}\sqrt{1+x^{2}}}$$ with a subsitution like this $x = \frac{1}{t}, t<0$ I'm on this step $$\int \frac{dt}{\frac{1}{t}\sqrt{t^{2} + 1}}$$ ...
2
votes
3answers
160 views

Not sure how to go about solving this integral

$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$ I've let $u=x^2+3$ but can't seem to get the right answer. Really not sure what to do.
11
votes
2answers
6k views

Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

How would you go about solving integral of a floor? The particular problem I have is: $$\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$$
147
votes
19answers
23k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
3
votes
3answers
583 views

Integrating $\int\frac{2dx}{x\ln(6x)}$

I needed some help integrating this: $$\int\frac{2\,dx}{x\ln(6x)}.$$ I have never seen the dx within the problem like that, I am assuming I can't just move it to the outside can I? Can I start by ...
8
votes
10answers
12k views

Evaluate $\int \frac{1}{\sin x\cos x} dx $

Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $ I know that the correct answer can be obtained by doing: $\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}...
93
votes
11answers
8k views

Ways to evaluate $\int \sec \theta \, \mathrm d \theta$

The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
11
votes
4answers
10k views

Integral of $\sqrt{1 + \sqrt{x}}$

My professor wants us to do this problem to refresh ourselves with substitution. We have to solve: $$\int\sqrt{1 + \sqrt{x}}\,\mathrm dx$$ $$\int\sqrt{1 + \sqrt{1 + \sqrt{x}}}\,\mathrm dx$$ ......