Definite integral Definition and 389 Threads

Calculation of Integral $\displaystyle \int_{0}^{1}x^{2014}\cdot \left(1-x\right)^{2014}dx$

Homework Statement I_1 = \int_0^{2\pi} \frac{sin\theta}{3+2cos\theta} d\theta Homework Equations Using identities to change from cos, sin, to variables of z, I get: 2iz^2 + 6iz + 2i in my denominator The Attempt at a Solution Looking for a singularity, will I use a quadratic...

Homework Statement $$ \displaystyle \int_0^{\infty} e^{-x} \dfrac{a\sin ax - \cos ax}{1+a^2} da $$ Homework Equations The Attempt at a Solution Evaluating this using integration by parts will be a cumbersome process and I don't even think that would give me the answer. Substitutions aren't...

Hello Can someone please tell me how is: \int_{-R}^{R} \frac{\cos mx}{x^2 + 1}\,dx = 2\int_{0}^R \frac{\cos mx}{x^2 + 1}\,dx where, m and R are positive real numbers This is how I'm trying to solve it... \int_{-R}^R \frac{\cos mx}{x^2 + 1}\,dx = \int_{-R}^0 \frac{\cos mx}{x^2 + 1}\,dx...

I am looking at a solution to an question and I don't understand how the value of the following definite integral comes out to be zero ? The following function is evaluated from 0 to ∞ with r being the variable ## exp(-β^2r^2)r^nr^-1/(n-1)## That should read r raised to the power of...

Question : https://www.physicsforums.com/attachments/71328 My question is how did the 2a and 2b come from?? Equations: Area of trapezoid =(a+b/2)(h) Attempt: I know that the area of a trapezoid is (a+b/2)(h) However why is there now a 2a and 2b in its place? Could it be related to the 2s...

Can anyone explain this to me? What does if mean that the area may sometimes be negative but that the area must be positive??

Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.

I know the value of the following definite integral \int_{a}^{b}ydx I also have a realtion x=f(y) i.e. x is an explicit function of y but I do not have y as an explicit function of x. The relation between x and y is generally non linear. Now I want to get the following definite...

Numerical integration methods applicable to a type of definite integrl Hey, so I've been working on a program to numerically integrate an integral of the form ∫xnf(x) dx, LIM(0 to INF.) Here n can go to negative non integral values, say -3.7 etc. and f(x) is a function of sin, cos and...

Is this possible.. Say, a∫b f(x)dx = G(x)|x=b - G(x)|x=a = S, where S, a and b are known. Can we find G(x) ?

Evaluation of $\displaystyle \int_{-5}^{-7}\ln \left(x-3\right)^2dx+2\int_{0}^{1}\ln(x+4)^2dx$ My Try:: Let $(x-3) = t$ Then $dx = dt$ and changing Limit, we get and Again put $(x+4) = u,$ Then $dx = du$ and changing Limit, we get $\displaystyle...

Evaluate $\displaystyle\int^{\dfrac{\pi}{4}}_0 \dfrac{x}{(\sin x+\cos x)\cos x}\ dx$.

Am trying to get a solution to the definite integral below. Looking for some direction. I = 0∫1 xf(x)dx where 0∫1 f(x)dx = A, is known. Also, its is know that when x =1, f(x) =0 and when x =0, f(x) =1. Can we get a solution of I in terms of A? I have tried going the...

☺'s question at Yahoo! Answers: a definite integral whose integrand has an absolute value factor. Here is the question: I have posted a link there to this thread so the OP can view my work.

Problem: Evaluate: $$\lim_{n\rightarrow \infty}\int_0^1 \frac{nx^{n-1}}{1+x}\,dx$$ Attempt: I used the series expansion: $$\frac{1}{1+x}=\sum_{r=0}^{\infty} (-1)^rx^r$$ From above, I got: $$\lim_{n\rightarrow \infty} \sum_{r=0}^{\infty} \frac{(-1)^rn}{n+r}$$ But I don't see how to proceed from...

Homework Statement Integrate:I=\int_{-π/4}^{π/4} \ln{(\sec θ-\tan θ)}\,dθ Homework Equations Properties of definite integrals, basic integration formulae, trigonometric identities. The Attempt at a Solution By properties of definite integrals, the same integral I wrote as...

I am trying to solve this integral: \int \frac{x^a-1}{ln(x)} dx (with the interval from 0 to 1). I have tried substitution but I could not find a way to get it to work. Any ideas on how to solve this? Thanks!

Problem: $$\int_0^{\infty} \frac{1}{x}\left(\frac{1}{1+e^x}-\frac{1}{1+e^{2x}}\right)\,dx$$ Attempt: I use the following two series expansions: $$\frac{1}{1+e^x}=\frac{e^{-x}}{1+e^{-x}}=e^{-x}\sum_{k=0}^{\infty} (-1)^k e^{-kx}=\sum_{k=0}^{\infty} (-1)^k e^{-(k+1)x}$$...

$\displaystyle \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 x}}dx = \int_{0}^{\frac{\pi}{4}}\frac{\pi}{2}-\int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{\cos 2x}{2\cos^2...

Homework Statement I'm trying to solve this definite integral using the residue theorem: \int _0^\pi \frac{d \theta}{ (2+ \cos \theta)^2} Homework Equations I got the residue theorem which says that \oint_C f(z)dz = 2 \pi i \ \ \text{times the sum of the residues inside C}...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Problem: Evaluate: $$\int_0^{\infty} t^{-1/4}e^{-t}\,dt$$ Attempt: I recognised this one as $\Gamma(3/4)$. I found a few formulas on Wolfram Mathworld website which helps to evaluate this but I am wondering if I can solve the definite integral from elementary methods (like by parts). Any help...

Problem: $$\int_1^e \frac{1+x^2\ln x}{x+x^2\ln x}\,\,dx$$ Attempt: I tried the substitution $\ln x=t \Rightarrow dx/x=dt$ and got the following integral: $$\int_0^1 \frac{1+e^{2t}t}{1+e^t t}\,dt$$ I am not sure how to proceed after this. :confused: Any help is appreciated. Thanks!

Evaluate: $$2^{2009}\frac{\displaystyle \int_0^1 x^{1004}(1-x)^{1004}\,dx}{\displaystyle \int_0^1x^{1004}(1-x^{2010})^{1004}\,dx}$$ ...of course without the use of beta or gamma functions. :p

Here is the question: I have posted a link there to this thread so the OP can view my work.

Evaluate the following: $$\int_0^{\pi} e^{\cos x} \cos(\sin x)\,\,dx$$

Problem: Find the value of $a$ such that $$\int_0^{\pi/2} |\cos(x)-ax^2|\,dx$$ is minimum. Attempt: Honestly, I don't know how to start. I tried the following: $$\int_0^{\pi/2} |\cos(x)-ax^2|\,dx \geq \int_0^{\pi/2}|\cos(x)|\,dx-\int_0^{\pi/2}|a|x^2\,dx=1-\frac{|a|\pi^3}{24}$$ $$\Rightarrow...

Attempt: If $\displaystyle U_n=\int_0^{\pi/2} \frac{\sin^2(nx)}{\sin^2x}\,dx$, then find $U_n-U_{n-1}$. Attempt: $$U_n=\int_0^{\pi/2} \frac{\sin^2(nx)}{\sin^2x}\,dx$$ $$U_{n-1}=\int_0^{\pi/2} \frac{\sin^2((n-1)x)}{\sin^2x}\,dx$$ $$\Rightarrow U_n-U_{n-1}=\int_0^{\pi/2}...

Problem: If f is continuous and differentiable function in $x \in (0,1)$ suuch that $\sum_{r=0}^{1}\left(f(x+r)-\left|e^x-r-1\right|\right)$=0, then $\int_0^{11} f(x)\,dx$ is A)65+4ln2-7e B)63+4ln2-9e C)69-9e D)29-23e Ans: A Attempt: I could only write the following...

Evaluate: $$\Large \int_{\pi/2}^{5\pi/2} \frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}$$

Compute: $$\int_0^{\pi/2} \tan(x)\ln(\sin(x))\,dx$$

Homework Statement 180\int_5^\propto \frac{2}{(4+x^{2})^{3/2}} dx Homework Equations Trigonometric Substitutions: (x=2 tan z). The Attempt at a Solution I've computed the integral and ended up with 180 [\frac{x}{2(4+x^2)^{1/2}}] from 5 to infinity. I could've easily computed...

My professor sent out an online work sheet with tons of practice problems, and I'm having trouble with this one, is my answer right? (see link) I chose this because a definite integral has to have limits, correct?

The problem is ∫x^2 - 3x - 5 with the lower limit being -4 and the upper limit 7. I broke the integrals into three parts from [-4, -1.1926], [-1.1926, 4.1926], [4.1926, 7] I did the integral and got (x^3)/3 - (3/2)x^2 - 5x I subbed in the lower and upper limits and got 32.861 for [-4...

Homework Statement Homework Equations solve the definite integral \int_{2.6}^{5.5} \frac{1}{x^2+9}dx The Attempt at a Solution ln(5.5^2+9)-ln(2.6^2+9) doesn't seem correct

Here is the question: I have posted a link there to this thread so the OP can view my work.

Here is the question: I have posted a link there to this thread so the OP can view my work.

I attached the solution from the solution manual of the integral I'm trying to figure out. \int_{-∞}^{∞}x^{2}exp(\frac{-2amx^{2}}{h}) The solution of that integral without the x2 in front is \sqrt{\frac{{\pi}h}{2am}} So with the x2 I assumed I needed to do integration by parts. So...

Hi, I'm wondering if I have the most direct solution for this integral or if there is a more efficient way of solving this. I haven't seen a double substitution deployed on one of these problems yet, so I thought perhaps this was not necessary. Homework Statement Using the substitution t...

Suppose $f(-x)=f(x)$, then compute the following definite integral: $$\int_{-a}^{a}\frac{1}{1+2^{f(x)}}\,dx$$ where $0<a\in\mathbb{R}$.

Here is the question: I have posted a link there to this thread so the OP can see my work.

Here is the question: I have posted a link there to this thread so the OP can see my work.

\int_0^{2\pi} \frac{1}{25cos^2(t) + 9sin^2(t)}dt Substituted the variables twice and got the upper and lower boundaries to both be 0 (think i might have gone wrong there) \frac{1}{15} tan^{-1} \frac{3tan(t)}{5} with upper and lower boundaries both being 0. I know the answer is 2\pi/15...

Homework Statement Let ##\displaystyle f(r)=\int_0^{\pi/2} x^r\sin x \,\, dx##. Now match the following List-I with List-II. $$ \begin{array} {|c| c | l c|} \hline & \text{List-I} & & \text{List-II} & & \\ \hline \text{(P)} & \lim_{r\rightarrow \infty}...

So the question is…Evaluate the following… $$\frac{d}{dx} \left(\int _1^{x^2} \cos(t^2) \, dt \right)$$ I thought i could use the FTC on this because it states… $$\frac{d}{dx} \left(\int_0^x f(t)\, dt \right)=f(x)$$ but i can't correct? because in my question it starts at 1…is there some way...

So the question is Evaluate (x-2)dx as the integral goes from -2 to 2 using the definition of a definite integral, choosing your sample points to be the right endpoints of the subintervals… Ok, so i understand how to do this problem if it gave me an actual number of interval like n=6 but it...

Calculation of : $\displaystyle \int_{0}^{1991}\{ \frac{2x+5}{x+1}\}[ x]dx$, where $[ x]$ and $\{ x \}$ denote the integral and fractional part of $x$ My Trial :: $\displaystyle \int_{0}^{1991}\left\{\frac{(2x+2)+3}{x+1}\right\}\cdot [x]dx$ $\displaystyle...

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement If the value of the integral ##\displaystyle \int_1^2 e^{x^2}\,\, dx## is ##\alpha##, then the value of ##\displaystyle \int_e^{e^4} \sqrt{\ln x} \,\, dx## is: A)##e^4-e-\alpha## B)##2e^4-e-\alpha## C)##2(e^4-e)-\alpha## D)##2e^4-1-\alpha## Homework Equations...