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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Evaluating $\int_\pi^{2\pi} \frac{x^2+2}{x^3}\cos x \,dx$

I'm trying to evaluate the following definite integral: $$\int_\pi^{2\pi} \frac{x^2+2}{x^3}\cos x \,dx$$ I tried integration by parts, the integral is getting only complicated with each step. Tried to apply definite integral properties, but its not…
Madavan Viswanathan
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Definite Integration with Inverse function

$$\int^{3}_{-1}\bigg(\tan^{-1}\bigg(\frac{x}{x^2+1}\bigg)+\tan^{-1}\bigg(\frac{x^2+1}{x}\bigg)\bigg)dx$$ what i try from…
jacky
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Calculating definite integral

I have this definite integral $$\int_{-1}^14x^3-x\,dx=0$$ I had that function rendered and found out it should be calculated in four intervals: $[-1,-0.5], [-0.5,0], [0,0.5], [0.5,1].$ Is there any other (shorter and correct) method how to…
naruto25
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Integrate the area beneath a circumference

I have got the following equation: $$(x-4)^2 + (y - 4)^2 = 16$$ I would like to find the area beneath this circumference between $x=\frac{8}{5}$ and $x = 4$ To do so, I would have to integrate $$(x-4)^2 + (y - 4)^2 = 16$$ how could I do that if…
Matheus Minguini
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Expand integral formula with differential dx

I read the following formula which expands the integral from SICP for small values of d x . We can express this directly as a procedure: (define (integral f a b dx) (define (add-dx x) (+ x dx)) (* (sum f (+ a (/ dx 2.0)) add-dx b) …
Wizard
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shorter or simpler evaluation of $\int_{-6}^6{\frac{19+20\sin^7x}{x^2+36}}\,\mathrm dx$

Are there any efficient ways to calculate this by hand? The integral appeared on a University engineering entrance exam to which I don't have the solutions. Putting it into an online integral calculator (https://www.integral-calculator.com/) gives…
Rob-bxl
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How to change limits of this integral to new ones

I wanna to change the limits of the following integral from $(0,0.6)$ to (-1,1), How can I do this? $$\int_0^{0.6} r^k e^{\sum_{n=1}^N \frac{c_n r^n}{(1+g*r)^{N-3}}}dr$$
Wisdom
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Calculate this integral containing log inverse tanh: $\int_0^1 x(\arctan x)\ln(\operatorname{arctanh}x)dx$

Can I get some help calculating the following integral $$I = \int_0^1 x(\arctan x)\ln(\operatorname{arctanh}x)dx$$ By using $\operatorname{arctanh}x=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ we…
tyobrien
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Finding value of $ \lim_{n\rightarrow \infty}\int^{1}_{0}x^{2019}\cdot \sin (nx)\,dx$

Finding value of $$ \lim_{n\rightarrow \infty}\int^{1}_{0}x^{2019}\cdot \sin (nx)\,dx$$ what i try $$I = \lim_{n\rightarrow \infty}\int^{1}_{0}x^{2019}\cdot \sin (nx)dx$$ Integration by parts $$I=\lim_{n\rightarrow \infty}\bigg[-x^{2019}\cdot…
jacky
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When is one allowed to merge "outside of integral" stuff inside integrals?

When is one allowed to merge "outside of integral" stuff inside integrals? Such as $$\int_A f(y)dy - f(x)$$ becoming $$\int_A [f(y)-f(x)] dy$$ Intuition says that in this case $f(x)$ doesn't depend on $y$, therefore "it doesn't alter then integral,…
mavavilj
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A definite integral involving trigonometric functions

Evaluate $$\alpha=\int_{\pi \over 6}^{\pi \over 3}\frac{\sin t+\cos t}{\sqrt{\sin 2t}}dt$$ My attempt: First I tried to substitute ${π \over 2}-x$, but didn't work . Then I made square in denominator $$\alpha=\int_{\pi \over 6}^{\pi \over…
Rishi
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How to solve the given multivariable integral using recursion relation?

This integral appeared in Statistical Mechanics while calculating the number of microstates for an ideal gas. $B_N=\int_{0}^{\infty}dy_1y_1^2\cdot\cdot\cdot\cdot\int_{0}^{\infty}dy_N y_N^2\Theta(1-\sum_{r=1}^{N}y_r)$ where $\Theta$ denotes the unit…
Asit Srivastava
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Evaluate $\int_0^{\pi/2} \cos^2(wt)dt$ where w is constant

We can proceed by $$\frac{\int_0^{\pi/2}(\cos2wt+1)dt}{2}$$ (I hope you don’t mind if don’t write the limits for the next few steps, as it is becoming cumbersome) So, $$\frac12\int \cos2wt+\frac12\int 1$$ $$\frac{1}{4w}\sin2wt+\frac t2$$ Putting…
Aditya
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Evaluating Definite Integrals with Primitive Functions

Evaluate: ∫_3^4▒(x^2+x+3)/(3x^5 ) dx =[(x^3/3+x^2/2+3x)/(x^6/2)] =(4^3/3+4^2/2+3(4))/(4^6/2)-(3^3/3+3^2/2+3(3))/(3^6/2) =31/1536-5/81 =-1723/41472 I need some help with the working for this definite integral question. Iv shown what I have done but…
user657464
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Estimating a Definite Integral

I have a problem asking me to show that $$\frac{3}{8}<\int_0^\frac{1}{2}\sqrt{\frac{1-x}{1+x}}dx <\frac{\sqrt3}{4}.$$ The left side of the equation is clear since $1-x<\sqrt{\frac{1-x}{1+x}}$ for $x \in (0,1/2)$. I cannot see a clean way to obtain…
user548941
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