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Yukti Vault

JEE Mathematics / Practice

Definite Integration

Chapter overview and full question list

Showing 358-362 of 392 questions.

IIT-JEE 1989Question 358
If $f$ and $g$ are continuous functions on $[0,a]$ satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then show that $\displaystyle\int_0^{a}f(x)g(x)\,dx=\int_0^{a}f(x)\,dx$.
Definite Integration

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IIT-JEE 1997Question 359
Let $a+b=4$, where $a<2$, and let $g(x)$ be a differentiable function. If $\dfrac{dg}{dx}>0$ for all $x$, prove that $\displaystyle\int_0^{a}g(x)\,dx+\int_0^{b}g(x)\,dx$ increases as $(b-a)$ increases.
Definite Integration

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IIT-JEE 1990Question 360
Show that $\displaystyle\int_{0}^{\pi/2}f(\sin2x)\sin x\,dx=\sqrt2\int_{0}^{\pi/4}f(\cos2x)\cos x\,dx$
Definite Integration

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IIT-JEE 1991Question 361
If $f$ is a continuous function with $\displaystyle\int_0^{x}f(t)\,dt\to\infty$ as $|x|\to\infty$, then show that every line $y=mx$ intersects the curve $y^{2}+\displaystyle\int_0^{x}f(t)\,dt=2$. [figure: curve through $A(0,\sqrt2)$ and $B(0,-\sqrt2)$ cutting the $x$-axis at $(x_p,0)$]
Definite Integration

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IIT-JEE 1990Question 362
Prove that for any positive integer $k$, $\dfrac{\sin2kx}{\sin x}=2\left[\cos x+\cos3x+\cdots+\cos(2k-1)x\right]$ Hence prove that $\displaystyle\int_{0}^{\pi/2}\sin2kx\,\cot x\,dx=\dfrac{\pi}{2}$
Definite Integration

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