Definite integral Definition and 389 Threads

For $$m \in \mathbb{Z}^+$$, and $$a, \, z \in \mathbb{R} > 0$$, evaluate the definite integral:$$\int_0^z\frac{x^m}{(a+\log x)}\,dx$$[I'll be adding a few generalized forms like this in the logarithmic integrals thread, in Maths Notes, shortly... (Heidy) ]

Hello, I would like to evaluate the following definite integral \int_0^1 \frac{exp(1/(x(1-x)))}{\sqrt{x(1-x)}} Numerically I get the result of about 1.4695 and it appears to converge nicely in the domain of interest (0;1). However, I'm wondering whether some kind of analytical integral...

Homework Statement The problem and my (incorrect) work are typed and attached as TheProblemAndMyWorkTypedUp.jpg. Homework Equations Integral from a to b of f(t) = F(b) – F(a) The Attempt at a Solution As mentioned above, my work is attached as TheProblemAndMyWorkTypedUp.jpg. (The (2 –...

Evaluate $$\int_0^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{\pi e}}$$.

Evaluate $$\int_0^{\pi} \frac{\cos 4x-\cos 4 \alpha}{\cos x-\cos \alpha} dx$$

i tried to solve this definite integral but i keep on getting an invalid answer. please check my error. $\displaystyle \int_{-3}^{-2}\frac{y+2}{y^2+4y}dy$ $\displaystyle u=y^2+4y$ $\displaystyle du=2y+4dy$ $\displaystyle dy=\frac{du}{2y+4}$ $\displaystyle \frac{1}{2}\int\frac{y+2}{u}\times...

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

Evaluate $$\int_2^4 \frac{\sqrt{\ln (9-x)}\,dx}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}$$

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

I want to proof \int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta [SIZE="5"](1) Let ##\theta=-\theta\;\Rightarrow\; d\theta=-d\theta,\;\cos(-\theta)=\cos\theta,\;\cos[m(-\theta)]=\cos(m\theta)## \int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta)...

Find the exact value of the following definite integral: $$\int_0^2 (3x^2-3x+1)\cos (x^3-3x^2+4x-2)\,dx$$

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

Hi, how are you? I came across some exercises that really puzzled me. They ask me to graph the following functions: $$ a) \int_0^x\sqrt{|tan(w)|} dw $$ $$ b)\int_0^\sqrt{x} e^{t^2}$$I imagine I'll have to use derivative techniques as I would when graphing a "normal" function, but those...

Homework Statement Given that x^{2}f(x)+f(\frac{1}{x})=0, then evaluate \int^{1.5}_{0.6}f(x)dx Homework Equations The Attempt at a Solution tried to replace f(x) using the provided equation...didn't help

Homework Statement If ##\displaystyle P=\int_0^{\pi} \frac{\cos x}{(x+4)^2}dx## and ##\displaystyle I=\int_0^{\pi/2} \frac{\sin (2x)}{2x+4}dx##, then the value of ##P+2I-\frac{1}{\pi+4}## is equal toHomework Equations The Attempt at a Solution By substituting 2x=t i.e 2dx=dt, and replacing t...

[FONT=Times]I figured I would just add this new problem over here, rather than starting a new thread. Im looking to solve integration leading to arctan or arcsin results. \int_{1}^{e}\frac{3dx}{x(1+\ln(x)^2}) Looking at this, it feels like this has an arctan in the result, but I would have to...

1. \int^{a}_{0} x\sqrt{x^{2}+a^{2}} a > 0 2. u = x^{2} + a^{2}, du = 2x 3. \frac{1}{2}\int^{a}_{0}\frac{2u^{3/2}}{3} = \frac{1}{3}\int ^{a}_{0}(x^{2}+a^{2})^{3/2} How do I solve this? Any hints?

Homework Statement Evaluate the definite integral ∫(x101-√(9-x^2))dx from -3 to 3. Hint: This problem can be done without anti-differentiation.Homework Equations The Attempt at a Solution I am stuck. I tried to do it with with anti-differentiation and it didn't work/very...

Homework Statement Let g be a continuous function on R that satisfies ##\displaystyle g(x)+2\int_{0}^{\pi/2} \sin x \cos t g(t)dt=\sin x##, then ##g'\left(\frac{\pi}{3}\right)## is equal to A)1/2 B)1/√2 C)1/4 D)none of these Homework Equations The Attempt at a Solution...

Homework Statement f(x) is a bounded function and integrable on [a,b] . a, b are real constants. We have to prove that i) An = a∫b f(x)cos(nx) dx → 0 when n→∞ ii)Bn = a∫b f(x)sin(nx) dx → 0 when n→∞ Homework Equations Parseval's formula : For uniform convergence of f(x) with its...

Homework Statement $$\int_{0}^{\ 2\pi} \ |e^{sin(x)}cos(x)| \, dx$$ I know that it simplifies to $$ 2e- \frac{2}{e} ≈ 4.7 $$ I'm not sure how to approach this problem. Do I just break the integral up into the domains where it's positive and negative and integrate each component...

Homework Statement \int_{0}^{2\pi} \frac{dx}{1+e^{\sin x}}Homework Equations The Attempt at a Solution Let ##I=\int_{0}^{2\pi} \frac{dx}{1+e^{\sin x}}## Since ##\int_{a}^{b}f(x) dx=\int_{a}^{b} f(a+b-x)dx## Hence, I=\int_{0}^{2\pi} \frac{dx}{1+e^{-\sin x}}=\int_{0}^{2\pi} \frac{e^{\sin...

Is there a closed form expression for the following definite integral? \int_{-∞}^{∞} exp(\frac{-|z|^2}{2{\sigma}^2}-\alpha |\mu + z|)dz where z is complex, and \alpha, \sigma, \mu are real constants. I couldn't obtain an expression similar to Gaussian integral, so I couldn't take the...

Here is the question: Here is a link to the question: Calculus question, please, please answer.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I have the following integral: \int_0^{f(x,y)}{f' \sin(y-f')df'} Now suppose that f(x,y) = x*y, my question is how do I write the integral in terms of x and y only? Can I do something like this? Since df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy we can obtain...

Homework Statement I=\int_0^{\pi} \frac{xdx}{9\cos^2 x+\sin^2 x} Homework Equations The Attempt at a Solution The given integral can be written as I=\int_0^{\pi} \frac{(\pi-x)dx}{9\cos^2 x+\sin^2 x} The denominator remains unchanged because ##\cos^2(\pi-x)=-\cos x## and square...

1. The problem, the whole problem, and nothing but the problem \int_0^\pi \frac{x \cdot sin(x)}{1+cos^2(x)} \, dx Homework Equations Integration by parts trig substitution The Attempt at a Solution My first idea was to break up the integral by letting u=x and dv=sin(x)/(1+cos^2 x). I will...

Homework Statement given that [SIZE="4"]∫01 xndx = 1/ n+1 for integers n \geq 0 calculate these integrals Homework Equations [SIZE="4"]∫01 (1+x+x2) dx The Attempt at a Solution I have found the definite integral of [SIZE="4"]∫01 (1+x+x2) dx to = 1.8333... (ie. 11/6) but all I...

Here is the question: Here is a link to the question: Calculus integral help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

According to wolfram alpha, the correct answer is -3/4.

Homework Statement Consider ##f(x)=4x^4-24x^3+31x^2+6x-8## be a polynomial function and ##\alpha, \beta, \gamma, \delta## are the roots of the equation ##f(x)=0##, where ##\alpha < \beta < \gamma < \delta##. Let sum of two roots of the equation f(x)=0 vanishes. Then the value of...

Here is the original question: Here is a link to the question: Ntegral of sin^3(6x)cos^4(3x) dx ? - Yahoo! Answers I have posted a link there to this topic so that the OP can find my response.

This is classic one. Prove that $$ \int_0^\infty \frac{dx}{\left\{x^4+(1+2\sqrt{2})x^2+1 \right\}\left\{x^{100}-x^{99}+x^{98}-\cdots +1\right\}}=\frac{\pi}{2(1+\sqrt{2})}$$

Homework Statement Evaluate: from x=0 to x=1 and y=0 to y=1 ∫(y^2 + 2xy(dy/dx))dx and carry the integration out over x Homework Equations The Attempt at a Solution I know how to calculate double integrals with multiple variables but the (dy/dx) throws me off and it says to...

Evaluate definite integral. "x if x<1; 1/x if x> or equal to 1." 1. Consider the function: f(x) = {x if x<1 {1/x if x≥1 Evaluate the definite integral. ∫from 0 to 4 of f(x)dx 2. Okay, I think I vaguely remember something about these...

Homework Statement Definite integral: \int \frac{\ln(x+1)}{x^2+1} dx from x=0 to x=1 (sorry I don't know how to do integral boundaries with tex) The Attempt at a Solution I just am clueless on how to do this, I'm almost 100% sure you are not supposed to find the anti-derivative...

Homework Statement Some friend of mine found this on a book: \int_{0}^{+inf}\frac{1-cos(\omega t)}{e^{\omega /C}(e^{\omega /T}-1)\omega }=ln[\frac{(\frac{T}{C})!}{|(\frac{T}{C}-iTt)!|}] The proof is left for the reader. Homework Equations The Attempt at a Solution First very safe step...

Prove that \[\int_0^1 \frac{\ln x}{\sqrt{x(1-x^2)}}dx=-\frac{\sqrt{2\pi}}{8} \left(\Gamma\left(\frac{1}{4} \right)\right)^2 \] \(\Gamma (x)\) is the Gamma Function.

I have a definite integral defined by \begin{equation}T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g\end{equation} where G is a continuous function of a variable g, and g_{1} and g_{2} are known numbers. I want to minimize T\left(G\left(g\right)\right), that is I want to...

Homework Statement Find the definite integral of a quarter circle. Homework Equations x^2 +y^2=10 The Attempt at a Solution x^2=10-y^2 x=sqrt(10-y^2) ∫ sqrt(10-y^2)dy from 0 to sqrt(10) I'm not sure what to do here.

Homework Statement ∫ 3+ x√x from -1 to 4 Homework Equations The Attempt at a Solution ∫ 3+ x√x from -1 to 4 = 3x+(2(x^5/2))/5 evaluated from -1 to 4 (12 + (2(4^5/2))/5) +(3 +(2(-1)^5/2)/5) ? 15+64/5 +(2(-1^(5/2))/5) 139/5 -(2(-1^(5/2)))/5 139/5 -2i/5 is that right?

I'm struggling for a long time to solve this integral $$\int_0^\infty e^{-x^2}cos(kx)dx$$ with $k>0$ I know there are a number of ways, but I'm interested in using complex integration. In particular, I believe that we can solve by integrating $e^{-z^2}$ over the boundary of the rectangule...

Homework Statement Express the following as a definite integral: Express the attached limit as an integral. The Attempt at a Solution I have gotten as far as every part of the answer except the upper bound. the answer is: 10 [SIZE="5"]∫(from 1 to 10) [x-4lnx]dx 1 since the...

Homework Statement I attached problem as a picture. Homework Equations I know that the integral of 1/x equals lnx if the derivative of the denominator is equal to the numerator. The Attempt at a Solution I tried to foil out the denominator and integrate by parts but it was very difficult...

$\displaystyle \int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx.\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}dx =$

Homework Statement Evaluate the definite integral from 0 to 18. ∫[x/(9+4x)^1/2]dx Homework Equations The Attempt at a Solution I know to use u-substitution and I set u = (9+4x)^1/2. I can't figure out where I'm messing up because I keep ending up with very large numbers which are...

$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x}{(a+b\cos x)^2}dx$

Hi, Suppose we have f(x) = x3. Integrating this function using the definite integral with the upper boundary being 3 and the lower boundary being 1 would result in 20. Does 20 have any units or is it unitless? Seeing how it is the area underneath a curve, I would imagine that it has square...

Homework Statement Three terms are used in a left hand sum to approximate the integral of ∫a to b f(x)dx ((2+0*(4/3))^2 * 4/3) + ((1+1*(4/3))^2 * 4/3) + ((2+2*(4/3))^2 * 4/3) find a possible value of b and a, and f(x Homework Equations Ʃ Δx(f(a+Δxi)) The Attempt at a Solution based...

Hi I would like to know what is the procedure to convert indefinite integral to definite one? For example I know ∫exp(-u^2)du from 0 to x is equal to ∫x*exp(-x^2*t^2)dt from 0 to 1 But I would like to know with what type of change of variable I get these?