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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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Integration of $\tan(\frac1x)$

$$\int_{-1}^{1}\tan\left(\frac1x\right) dx$$ How do I proceed? Please help.
Najmul
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Finding value of $\int^{\pi}_{0}\frac{\sin^2(10x)}{\sin^2 (x)}dx$

Finding value of $$\int^{\pi}_{0}\frac{\sin^2(10x)}{\sin^2 (x)}dx$$ Try: Using $\displaystyle \sin(10x)=\frac{e^{i(10x)}-e^{-i(10x)}}{2i}$ and $\displaystyle \sin (x)=\frac{e^{i(x)}-e^{-i(x)}}{2i}$ So $$\frac{\sin(10x)}{\sin x}=e^{-i(9x)}\cdot…
DXT
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Value of product of cosines definite integration

If $$I_{m}=\int^{2\pi}_{0}\cos x \cos (2x) \cos (3x)\cdots\cos(mx) \, dx.$$Then $m$ for which $I_m \neq 0$ is, where $m$ is any natural number Options $(a)\; 5\;\; (b)\; 6\;\; (c)\; 7\;\; (d)\; 8$ Try: $$I_m = \frac 1 {2^m}\int^{2\pi}_0…
DXT
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Confirmation of definite integral [0, 2 pi] of exp[ r cos(x)+ s sin(x)] cos[a cos(x)+b sin(x)]

I found in a paper the following integral form: $ \int_{0}^{2 \pi} exp[(r \cos(x)+s \sin(x)]\cdot \cos[( a \cos(x)+b \sin(x)] dx $ $= \pi[I_0(\sqrt{C+iD})+I_0(\sqrt{C-iD})]$ with $I_0$ as modified Bessel function, $i$ as imaginary unit and the two…
R. Schmidt
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The Green Book of Math Problems Question 5

Evaluate the following integral where it is noted that the denominator never vanishes over the interval. No assumption is made on the continuity of f(x) or its derivatives. $$\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} dx$$
donal obaoill
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Finding relation between $A$ and $B$ is

If $\displaystyle A=\int^{x}_{0}e^{zx}e^{-z^2}dz$ and If $\displaystyle B=\int^{x}_{0}e^{-\frac{z^2}{4}}dz$ Then relation between $A$ and $B$ is Try: assuming…
DXT
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Trigonometric exponential Integration

$$\int_{0}^{\frac{\pi}{4}}\frac{e^{\sec x}\cdot \sin \bigg(x+\frac{\pi}{4}\bigg)}{(1-\sin x)\cdot \cos x}dx$$ Try: substitute $\displaystyle x+\frac{\pi}{4}=t$ then $dx=dt$ So…
DXT
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Definite integral $\int _{-1}^1\:\frac{\arccos\left(x\right)}{1+x^2}dx$

Do you have any idea about how should I approach this integral? I tried various substitutions and ended up nowhere. $$\int _{-1}^1\:\frac{\arccos\left(x\right)}{1+x^2}dx$$
user528021
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Does converting an integral to multiple integrals always make the integral easier to solve?

Recently I've tackled the Putnam 2005 A5 integral, that is, $\displaystyle\int_{0}^{1}\frac{\ln(x+1)}{x^2+1}\mathrm{d}x$. In solving this, I converted it to multiple integrals- I used a $u$ substitution to show it was equal to…
Maxim Enis
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Compute the following double integral

Let $$I = \iint_D \frac{x^2+2x}{\sqrt{y}}\,dA,$$ where $D = \{y = \frac{1}{x}, y=\frac{2}{x}, x=y, y=\frac{x}{2}, x>0\}$ I graphed the functions from $D$, and the graph looked like this: graph So from this graph I can see that $D = D_1 \bigcup D_2$,…
Raducu Mihai
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Finding $\int^{\pi}_{0}x^4\sin^4(x)dx$

Finding $\displaystyle \int^{\pi}_{0}x^4\sin^4(x)dx$ Try: $$I= \int^{\pi}_{0}x^4\sin^4(x)dx= \int^{\pi}_{0}(\pi-x)^4\sin^4(x)dx$$ $$I=\int^{\pi}_{0}\bigg(\pi^4-4\pi^3x+6\pi^2x^2-4\pi x+x^4\bigg)\sin^4 xdx$$ Could some help me to solve it, Thanks
DXT
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Finding value of $k$ in definite Integral

If $\displaystyle \int^{\pi}_{0}\ln(1-2a\cos x+a^2)dx=k\ln(a),a>0$ Try: assume $$I(a)=\int^{\pi}_{0}\ln(1-2a\cos x+a^2)dx=k\ln(a)$$ Differentiate both side with respect to $a$ $$I'(a)=\int^{\pi}_{0}\frac{2a-2\cos x}{1-2a\cos…
DXT
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Area between curves finding pressure

The region |$y|>3|x|$ for $0
Shreyas Hegde
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Find definite integral based on other definite integrals

Let $\int^4_{-2}f(x)dx = 3$, $\int^0_{-2}f(x)dx=10$, $\int^4_2f(x)dx=2$ Find $\int^2_0f(x)dx = -9$ and $\int^0_2(3f(x) - 10)dx=??$ I figured out the first part is -9 since 10+2-9=3. I thought the second part would be $3(-9)-10=-37$ since f(x) from 0…
Jrow
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Use part 1 of the Fundamental Theorem of Calculus to find the derivative of the following function.

$$y = \int_\sqrt x^{\pi/4} \theta \tan\theta \, d\theta$$ I'm using the property of definite integrals that says $\int_b^a f(x) \, dx = -\int_a^b f(x) \, dx$ and I'm getting $y'= -\sqrt x \tan\sqrt x $ but the answer is $y'= -1/2 \tan\sqrt x…
Katelyn
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