Mathematics Stack Exchange
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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.

20559 questions
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How to evaluate $\int_{0}^{\pi }\frac{x\sin x}{1+\cos^{2}x}dx$

How can I evaluate this integral? $$\int_{0}^{\pi }\frac{x\sin x}{1+\cos^{2}x}dx$$
g3d
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How to evaluate $\int_0^z(x(z-x))^ndx$

How to evaluate $\int_0^z(x(z-x))^ndx$ I was trying to find conditional distribution involving two Gamma Distributions when this integral came up. Accoring to https://www.integral-calculator.com/ the answer is $B(n+1,n+1)z^{2n+1}$ I'm not sure how…
Anvit
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Complex logarithmic integral having Trigonometric function

Finding $\displaystyle \int^{2\pi}_{0}\ln|2\sin(x)+1|dx$ This is what I've tried: Let $\displaystyle I =\int^{2\pi}_{0}\ln|2\sin(x)+1|dx=\int^{2\pi}_{0}\ln|-2\sin x+1|dx$ $\displaystyle…
jacky
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$\int_0^1\frac{\ln{x}\ln{(1+x)}}{1+x}dx$

I want to solve for the following Integral: $$\int_0^1\frac{\ln{x}\ln{(1+x)}}{1+x}dx$$ I have tried to use: $$\ln{(1+x)}=-\sum_{k=1}^\infty\frac{(-1)^kx^k}{k}$$ and…
Reynan Henry
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How can this integral be solved?

I am trying to solve (by hand) the integral:$$\int_0^∞\frac1x\left(\frac{(x+a)^2}{((x+a)^2+b)^{3/2}}-\frac{(x-a)^2}{((x-a)^2+b)^{3/2}}\right)\,\mathrm dx,$$ where $a$ and $b$ are real and positive. Mathematica actually generates the answer, but I am…
WillG
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calculation of $\int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x^2+3x+2)}dx$

Calculation of $$\int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x^2+3x+2)}dx$$ My Try: Let $$I = \int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x+1)(x+2)}dx$$ Put $x=\sin^2 \theta$ and $dx = 2\sin \theta \cos \theta$ and changing limits So $$I =…
DXT
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If $ f(x) = x^3+\frac{3x}{4}-\frac{3x^2}{2}+\frac{7}{8}.$ Then $ \int^{\frac{3}{4}}_{\frac{1}{4}}f(f(x))dx$

If $\displaystyle f(x) = x^3+\frac{3x}{4}-\frac{3x^2}{2}+\frac{7}{8}.$ Then $\displaystyle \int^{\frac{3}{4}}_{\frac{1}{4}}f(f(x))dx$ Try: Given $\displaystyle f(x) =x^3+\frac{3x}{4}-\frac{3x^2}{2}+\frac{7}{8}. $ Then $\displaystyle f(1-x) =…
DXT
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How to do integral $ \int_0^{\infty} \left(\frac{a}{(e^{ax}-1)}-\frac{b}{(e^{bx}-1)}\right)\mathrm{d}x$?

I met this integral: $$ \int_0^{\infty} \left(\frac{a}{(e^{ax}-1)}-\frac{b}{(e^{bx}-1)}\right)\mathrm{d}x, \text{ where } \,\,0
unicornki
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Integrating with a max function as denominator.

I am trying to integrate forces over an area for a personal project of mine. I simplified and extracted the integrals that i could, but i ended with these: $$0.5 \int_0^1 \frac{x}{max(x(P_{1y} - P_{2y}) + P_{2y} - w, 0) + h} dx$$ that i try to…
Wicpar
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Finding the value of integral.

If $ \int_{-\infty}^{\infty} f(x) dx= 1,$ Find value of $ \int_{-\infty}^{\infty} f(x-\frac{1}{x}) dx$. I tried substituting $x$ as $\frac{1}{t}$, but nothing is happening. In the denominator, $1+x^2$ is left.
Math_centric
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minimum value of $I(l)$ in definite integration

If $\displaystyle I(l)=\int^{\infty}_{0}\frac{x^l}{2x^6+4x^5+3x^4+5x^3+3x^2+4x+2}dx.$ for $0
jacky
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Evaluate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^3 x\cdot f(\cos x){\rm d}x$

Probem Evaluate $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^3 x\cdot f(\cos x){\rm d}x.$$ Thanks to @NewBornMATH's hint. It's easy to verify that the integrand is an odd function，and the integral interval is symmetric with respect to the original…
mengdie1982
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How to evaluate the following integrals?

This is a problem of ordinary calculus. Given \begin{align} f(x)&= \left\{ \begin{array}{ccc} k(0),&\quad 0\leq x
Jay
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Find the value of a such that the area bound between the curves $e^{(ax^2)}$ and $e^{1/8}$ and the lines x = 0 and x= 1 is minimum

Find the value of a such that the area bound between the curves $e^{(ax^2)}$ and $e^{1/8}$ and the lines x = 0 and x= 1 is minimum I found out the point of intersection $\frac{1}{2 \sqrt{2a}}$ then found out the area after that used Newton lebnitz…
Avyansh Katiyar
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$\int_0^z a^x b^{x^{m+1}} dx$

Let $a$ and $b$ be positive constants, $m$ be any non-negative constant , how do we evaluate the following integral: $$\int_0^z a^x b^{x^{m+1}} dx$$ (at least a good approximation would do)
Souvik Dey
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