The Stack Overflow podcast is back! Listen to an interview with our new CEO.

# Questions tagged [indefinite-integrals]

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

386 questions
Filter by
Sorted by
Tagged with
9answers
2k views

### Indefinite integral of secant cubed

I need to calculate the following indefinite integral: $$I=\int \frac{1}{\cos^3(x)}dx$$ I know what the result is (from Mathematica): $$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$ but I don't ...
19answers
24k 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 ...
11answers
9k 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 ...
6answers
48k views

### Evaluating the indefinite integral $\int \sqrt{\tan x} ~ \mathrm{d}{x}.$

I have been having extreme difficulties with this integral. I would appreciate any and all help. $$\int \sqrt{\tan x} ~ \mathrm{d}{x}.$$
2answers
24k views

### Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
20answers
796 views

### Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ [closed]

I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
8answers
8k views

### Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as ...
3answers
94k 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 ...
4answers
525 views

### Evaluating this integral $\small\int \frac {x^2 dx} {(x\sin x+\cos x)^2}$

The question: Compute$$\int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2}$$ Tried integration by parts. That didn't work. How do I proceed?
3answers
3k views

### What is the integration of $\int 1/(x^{2n} +1)dx$?

I am a student who is preparing for IIT exam. I was just practicing calculus and encountered this problem. I tried different substitutions but none of them seemed to work. So what is the integration ...
15answers
29k views

### Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
3answers
4k views

### $-1 = 0$ by integration by parts of $\tan(x)$

I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation. A friend of mine forgot that such a primitive could easily be ...
3answers
642 views

### Solving this integral $\int\frac{1}{1+x^n} dx$?

Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned} Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z ...
11answers
16k views

### Demystify integration of $\int \frac{1}{x} \mathrm dx$

I've learned in my analysis class, that $$\int \frac{1}{x} \mathrm dx = \ln(x).$$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without ...
3answers
1k views

6answers
10k views

### What is the antiderivative of $e^{-x^2}$

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I ...
2answers
9k views

### When the integral of products is the product of integrals.

I'm self-studying and was doing the following integral: $$I = \int \frac{e^{\frac{1}{x}+\tan^{-1}x}}{x^2+x^4} dx$$ I solved it fine by letting $u = \frac{1}{x} + \tan^{-1}x$. My question is ...
4answers
7k views

### Evaluate $\int \cos(\cos x)~dx$

Evaluate $\int \cos(\cos x)~dx$ I tried to use chain rule but failed. Can anyone help me please?
3answers
5k views

### Integral $\int\!\sqrt{\cot x}\,dx$

Find the integral $$\int\!\sqrt{\cot x}\,dx$$ How can one solve this using substitution? Can this be solved by complex methods?
3answers
20k views

2answers
2k views

### Families of functions closed under integration

What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$? Here are ...
2answers
3k views

### Need help solving - $\int (\sin 101x) \cdot\sin^{99}x\,dx$

I have a complicated integral to solve. I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts. are there any simpler methods to evaluate this ...
4answers
14k views

5answers
1k views