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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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convergences of improper irrational Integral

Finding Convergence or Divergence of $$\int^{\infty}_{0}\frac{1}{\sqrt{x^6+1}}dx$$ What i Try: I am trying to prove $(x^2+1)\leq \sqrt{x^6+1}$ for all $x\geq 0$ $(x^2+1)^2\leq (x^6+1)\Longrightarrow x^4+2x^2+1\leq x^6+1$ Getting $x^4+2x^2\leq x^6$…
jacky
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A trigonometric definite integral with a parameter $\int_0^\pi\frac{dx}{1+\alpha^2\sin^2(x)}$

Give the expression of the integral $$\int_0^\pi\frac{dx}{1+\alpha^2\sin^2(x)}$$ where $ \alpha \in (0,+\infty)$. I tried the substitution $ t=\tan(x)$ but both bounds become zero. I used $t=\tan(\frac x2) $ but it became complicate, i think there…
hamam_Abdallah
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Area between triangle and parabola

Consider a triangle of vertices $(-3,2),(1,4),(3,1)$ Describe the area of region between the $y=x^2$ and inside of the triangle mentioned above as a definite integral. Can someone help me solve this problem.
Hitman
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values of $a$ for which the integral $\int^{\infty}_{0}e^{-at}\sin(7t)dt$ converges

Find values of $a$ for which the integral $$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$ converges What i try $$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$ $$=\frac{1}{a^2+49}\bigg(-e^{-at}a\sin(7t)-7e^{-at}\cos(7t\bigg)\bigg|^{\infty}_{0}=\frac{7}{a^2+49}$$ The…
jacky
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How many terms are needed to approximate wifhinv$0.01$

In a series expansion of given definite Integration $$\int^{1}_{0}e^{-x^2}dx$$ How many terms of this series are necessary to approximate this integral to within $0.01$ What i…
jacky
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finding real $k$ for which the definite Integral Converges

Evaluation of convergence of $$\int^\infty_1\frac{\log(1+x)}{x^k}dx$$ Where $k\in(0,\infty)$ What i try Using Integration by parts $$I=\ln(1+x)\cdot \frac{x^{1-k}}{1-k}\bigg|^{\infty}_{1}-\int^{\infty}_{1}\frac{1}{1+x}\cdot…
jacky
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fundamental theorem of integration to find definite Integration

Using fundamental Theorem of calculus to evaluate $$\int^{15}_{6}(6-x)(x-15)dx$$ What i try: $$I =\int^{15}_{6}(6-x)(x-15)dx$$ $$I =\int^{15}_{6}(21x-x^2-90)dx=\bigg(\frac{21}{2}x^2-\frac{x^3}{3}-90x\bigg)\bigg|^{15}_{6}$$ I did not understand why…
jacky
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Graph of $\sqrt x+\sqrt y=1$

The area (in sq. units) of the region bounded by the curve $\sqrt x+\sqrt y=1,x,y\ge0$, and the tangent to it at the point $(\frac14,\frac14)$ is : $\frac1{24}/\frac1{8}/\frac1{36}/\frac1{12}$? To start this, I need to make graph of $\sqrt x+\sqrt…
aarbee
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Finding sum : $\int_{0}^{\frac{\pi}{2}} ( 4\sin x-2\cos(2x)+2)^{\frac{1}{3}} \cos x dx+\int_{0}^{\frac{\pi}{2} }\sqrt { 1 + 8 \sin^3 x} \cos x dx$

$$ \int_{0}^{\frac{\pi}{2} }\sqrt { 1 + 8 \sin^3 x} \cos x dx = I_{1}$$ $$ \int_{0}^{\frac{\pi}{2}} ( 4\sin x-2\cos(2x)+2)^{\frac{1}{3}} \cos x dx= I_2$$ Find $I_1 + I_2$ My attempt: I converted this into a polynomial integral as $ -\int_{0}^{1}…
tryst with freedom
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Definite Integrals: How to get from a certain step to answer

I was working on a problem $\int^1_0$8sin(5$\pi$x)dx earlier and was shown the solution on the site I'm using to complete this work. I'd like an explanation on how, at a later point in my process/the way the site shows it,…
Rachel
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value of $f(x)$ for which definite integral has minimum value.

Find all polynomial $f(x)$ of degree $\leq 3$ for which $$\int^{\pi}_{-\pi}\bigg(\cos(x)-f(x)\bigg)^2dx$$ has minimum value. What i try: Let $f(x)=px^3+qx^2+rx+s$. Then $$\int^{\pi}_{-\pi}\left(f(x)^2-2f(x)\cos(x)+\cos^2(x)\right)dx.$$ Here, I…
jacky
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value of $k$ for which Definite Integration has finite value

Finding Value of $k$ for which $$\int^{1}_{0}\frac{\ln(x)}{x^k}$$ have finite value. What I tried: Put $\ln(x)=u$ and $x=e^u$ and $dx=e^udu$ and changing limits $$I =\int^{0}_{-\infty}u\cdot e^{u(1-k)}du$$ $$I =\frac{1}{1-k}\biggl(u\cdot…
jacky
  • 5,194
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Determine the value of a limit involving an integral

For any natural number $ a $, determine the value of $$ \lim_{n \to \infty} n \int_1^e x^a (\log (x))^n \,dx.$$ How we to find the value of the integral value? Please help me. Thank you in advance.
Brendon S
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Question about integral properties

Been stuck on this one for a while now. Any help would be appreciated: Let $f:[a,b]\rightarrow\mathbb R$ be an integrable function. Prove that there exist a $c\in[a,b]$ such that: $$\int_a^c f(x)dx=\int_c^b f(x)dx$$
Orokusaki
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Solving the integral without using Laplace

On my last question, i was trying to find the Laplace of the following function, $$\int_0^\infty \frac{e^{-2t}\sinh t\sin t }{t} dt.$$ Now I am wondering, can I solve the integral directly without using the Laplace, could someone provide me some…
kiv
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