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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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A problem on Riemann-Stieltjes integral

Compute the following Riemann-Stieltjes integral $\int_{-1}^{1} f(x)dg(x)$ where $f(x)=x^2 + e^x$ and $g(x)=Sgn(x)$. I found the answer is 2 but answer given is 1. I have used the following formula $\int_{a}^{b} f(x)dg(x)$ + $\int_{a}^{b}…
Kousik Sett
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A Variable representing an area in a definite integral

The integral in question The homework is already turned in so I won't be getting any credit changing it now but my instructor is adamant about the answer to part B. I just want some confirmation and justification as to what the correct answer is. As…
000
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Evaluating the greatest integer value of integral

If $$\int_{0}^{\infty}\frac{\ln x dx}{x^2+e^2}=k$$ then what is $[k]$=? The thing is I cant evaluate even 'k'! Please help me with this. Thanks in advance.
user323082
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Find $\int_0^{\frac{\pi}{4}}(\cos2x)^{3/2}\cos x dx$

Find $$\int_0^{\frac{\pi}{4}}(\cos2x)^{3/2}\cos x dx$$ My attempt: $$I=\int_0^{\frac{\pi}{4}}(\cos2x)^{3/2}\cos x dx=\int_0^{\frac{\pi}{4}}(1-2\sin^2x)^{3/2}\cos x dx$$ Let $\sin x=t$ $$I=\int_0^{\frac{1}{\sqrt2}}(1-2t^2)^{3/2}dt$$ I am stuck here.
Brahmagupta
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how to solve the following definite integral?

I am really confused about solving it. $$\int_{-2}^2 \frac{x^2}{1+5^x} \, dx $$
Gautam Chandna
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Is it true that $\int_b^ax^ndx=\int_b^{\infty}x^ndx+\int_{\infty}^ax^ndx$?

Messing around with the idea of point convergence, I found out that the following (principal value of the) integral can be evaluated as follows: $$\int_b^ax^ndx=\int_b^{+\infty}x^ndx+\int_{-\infty}^ax^ndx$$ You may observe its truth, graphically,…
Simply Beautiful Art
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Trouble evaluating an integral

I'm trying to evaluate the expression: $\mathbb{E}_h\{h^{0.5}\gamma(0.5, shc)\}$ where, $\gamma(.)$ is a lower incomplete gamma function given by $\gamma(a, z) = \int_{0}^{z}exp(-t)t^{a-1}dt$ and $h$ is an exponential random variable. A couple of…
ariadnus
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A simple Integral Question

What are the steps to calculate the value of $c$ in the following integral equation? $$ \int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}c.e^{-(x_1+2x_2+3x_3)}\,dx_1 \, dx_2 \, dx_3 = 1 $$
AKH
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An infinite integral of cos(x^3).

Is there any results about calculating this integral?? $f(\lambda )=\int_{-\infty }^{+\infty } \cos (x^3) \cos(\lambda x) \mathrm{d} x, \lambda >0.$
Kirby Lee
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How would I solve $\int \left(34^3-5t^{-3}\right)dt$?

I stumbled upon this section of my textbook and I don't know where to begin. We haven't reached this section in class yet but I wanted to know how to solve stuff like this.
Marygrace
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Relation Beta function, hypergeometric and trigonometric functions

Is there a simple reason, why the integrals of the form $$ C_1(s) =\int\limits_{0}^{\pi/2} \cos(2 \nu \theta) \cos^{2s-1}(\theta) \; d \theta $$ and $$ C_2(s)= \int\limits_{0}^{\infty} \cosh(2 \nu \theta) \cosh^{-1-2s}(\theta) \; d \theta,$$ can…
Marc Palm
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The Interpretation of Henstock Integral

If riemann integral has interpretation as the area under function, then what the interpretation of henstock integral? I always think bout it but don't get it
Albert Parulian
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Proof that $\int _{ 0 }^{ \pi/2 }{f(\sin2x)\sin x dx } $=$\sqrt2\int _{ 0 }^{ \pi/4}{f(\cos2x)\cos x dx }$

How would you prove $\int _{ 0 }^{ \pi/2 }{f(\sin2x)\sin x dx } $=$\sqrt2\int _{ 0 }^{ \pi/4}{f(\cos2x)\cos x dx }$ ? Its a problem from IIT-JEE 1990. I've tried it but i'm just going round and round. Give your suggestions please.
user220382
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Definite integral related to even function property

If $\displaystyle f\left( x \right)=\int _{ 0 }^{ x }{ g\left( t \right) dt }$,where g is an even function and $f(x+5)=g(x)$ then how to prove the following $$g(0)-g(x)=\int _{ 0 }^{ x }{ f\left( t \right) dt }$$
user220382
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Estimates of $f(x) = \int_x^{x^2} \dfrac{dy}{\ln y}$

I'm trying to solve this question: Let $f$ be the function defined for $x \in (0;1): f(x) = \int_x^{x^2} \dfrac{1}{\ln y}\,dy$. Show that $$\forall x \in (0;1), x^2 \ln2 \le \ f(x) \leq x \ln2$$ Probably it's nothing difficult, but I'm completely…
kilpikonna
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