Question:
$\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left[\left(x+\pi\right)^{3}+cos^{2} \left(x+3\pi\right)\right]dx$ is equal to :
$\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left[\left(x+\pi\right)^{3}+cos^{2} \left(x+3\pi\right)\right]dx$ is equal to :
Updated On: Jul 27, 2022
- $\left(\frac{\pi^{4}}{32}\right)+\left(\frac{\pi}{2}\right)$
- $\frac{\pi}{2}$
- $\left(\frac{\pi}{4}\right)-1$
- $\frac{\pi^{4}}{32}$
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The Correct Option is B
Solution and Explanation
Let $\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left[\left(x+\pi\right)^{3}+cos^{2} \left(x+3\pi\right)\right]dx ...(i)$
and $I=\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left[\left(-\frac{\pi}{2}-\frac{3\pi}{2}-x+\pi\right)^{3}+cos^{2}\left(-\frac{\pi}{2}-\frac{3\pi}{2}-x+3\pi\right)\right]dx$
$\Rightarrow $ $I=\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left[-\left(x+\pi\right)^{3}+cos^{2}\left(\pi-x\right)\right]dx ...\left(ii\right)$
On adding Eqs. (i) and (ii), we get
$2I=\int\limits^{-\pi/2}_{{-3\pi/2}}$$2\,cos^{2}\,x\,dx$
$=\int\limits^{-\pi/2}_{{-3\pi/2}}$$\left(1+cos\,2x\right)dx$
$=\left[x+\frac{sin\,2x}{2}\right]^{-\pi/2}_{_{_{-3\pi/2}}}$
$=-\frac{\pi}{2}+\frac{3\pi}{2}=\pi$
$\Rightarrow I=\frac{\pi}{2}$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.