Evaluate the following definite integrals: _ 0 ^2 ^ x ( 4 + x 2 )… - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{2\pi}\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{2\pi}\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{dx}$ Then,
Integrating by parts,
$\text{I}=\Big[2\text{e}^{\text{x}}\sin\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\Big]^{2\pi}_0-\int_{0}^\limits{2\pi}2\text{e}^{\text{x}}\sin\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{dx}$
Integrating second term by parts,
$\text{I}=\Big[2\text{e}^{\text{x}}\sin\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\Big]^{2\pi}_0+\bigg\{\Big[4\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\Big]^{2\pi}_0\\+\int_{0}^\limits{2\pi}-4\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{ dx}\bigg\}$
$\Rightarrow\text{I}=\Big[2\text{e}^{\text{x}}\sin\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\Big]^{2\pi}_0+\Big[4\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\Big]^{2\pi}_0-4\text{I}$
$\Rightarrow5\text{I}=-2\text{e}^{2\pi}\frac{1}{\sqrt{2}}-2\frac{1}{\sqrt{2}}-4\text{e}^{2\pi}\frac{1}{\sqrt{2}}-4\frac{1}{\sqrt{2}}$
$\Rightarrow5\text{I}=-3\sqrt{2}\text{e}^{2\pi}-3\sqrt{2}$
$\Rightarrow\text{I}=-\frac{3\sqrt{2}}{5}\big(\text{e}^{2\pi}+1\big)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Integrate the function $\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}$ with respect to $x$.

Let $A = R_0 \times R,$ where $R_0$ denote the set of all non$-$zero real numbers. A binary operation $'⊙\ '$ is defined on $A$ as follows: $(a, b) ⊙ (c, d) = (ac, bc + d)$ for all $(a, b), (c, d) \in R_0 \times R.$ Find the invertible elements in $A.$

Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies.

There are two bags $I$ and $II.$ Bag I contains $3$ white and $4$ red balls and Bag $II$ contains $5$ white and $6$ red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag $II.$

Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}'(\text{x})=\frac{\text{x}}{2}+\frac{2}{\text{x}}, \text{x}\geq0$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x+y}}+\text{e}^{-\text{x+y}}$

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=11\hat{\text{i}},\vec{\text{b}}=2\hat{\text{j}},\vec{\text{c}}=13\hat{\text{k}}$

Determine P(E|F) in Exercises.
Two coins are tossed once, where
E : no tail appears, F : no head appears.

Evaluate the following integrals:
$\int\cos^5\text{x}\text{ dx}$

Show that $f: [–1, 1] \rightarrow R$, given by $f(\text{x})=\frac{\text{x}}{(\text{x}+2)}$ is one-one. Find the inverse of the function $f:[-1, 1] \rightarrow $ Range $f.$
(Hint: For $\text{y}\in\text{Range f},\text{y}=f(\text{x})=\frac{\text{x}}{\text{x}+2},$ for some $x$ in $[-1, 1],$ i.e., $\text{x}=\frac{2\text{y}}{(1-\text{y})}$)