Evaluate the following definite integrals: _ 0 ^1 (xe^ x + 4 )dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$

✓

Answer

We have,
$\int_{0}^\limits{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$
$=\int_{0}^\limits{1}\text{xe}^{\text{x}}\text{ dx}+\int_{0}^\limits{1}\cos\frac{\pi\text{x}}{4}\text{ dx}$
Applying by parts in $1^{st}$ integral we get,
$=\text{x}\int_{0}^\limits{1}\text{e}^{\text{x}}\text{ dx}-\int_{0}^\limits{1}\big(\int\text{e}^{\text{x}}\text{ dx}\big)+\int_{0}^\limits{1}\cos\frac{\pi\text{x}}{4}\text{ dx}$
$=\big[\text{xe}^{\text{x}}\big]^1_0-\int_{0}^\limits{1}\text{e}^{\text{x}}\text{ dx}+\bigg[\frac{\sin\frac{\pi\text{x}}{4}}{\frac{\pi}{4}}\bigg]^1_0$
$=\big[\text{xe}^{\text{x}}-\text{e}^{\text{x}}\big]^1_0+\frac{4}{\pi}\Big[\frac{1}{\sqrt{2}}-0\Big]$
$=\big[\text{e}^{\text{x}}(\text{x}-1)\big]^1_0+\frac{4}{\pi}\Big[\frac{1}{\sqrt{2}}\Big]$
$=0+1+\frac{4}{\pi\sqrt{2}}$
$=1+\frac{2\sqrt{2}}{\pi}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs. 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs. 80. How many items of each product should be produced by the company so that the profit is maximum?

An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, $50\%$ are manufactured on machine A, $30\%$ on B and $20\%$ on C, $2\%$ of the items produced on A and $2\%$ of items produced on B are defective and $3\%$ of these produced on C are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that,

Both balls are red,
First ball is black and second is red,
One of them is black and other is red.

Show that the plane vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=1$ and the line whose vector equation is $\vec{\text{r}}=(-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})$ are parallel. Also, find the distance between them.

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their area is least when the side of square is double the radius of the circle.

Find the equation of the plane passing through the points whose coordinates are $(-1, 1, 1)$ and $(1, -1, 1)$ and perpendicular to the plane $x + 2y + 2z = 5.$

Probabilities of solving a specific problem independently by A and B are $\frac{1}{2} \text{and}\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that(i) the problem is solved (ii) exactly one of them solves the problem.

Maximise Z = x + y subject to $\text{x}+4\text{y}\leq8,2\text{x}+3\text{y}\leq12,3\text{x}+\text{y}\leq9,\text{x}\geq0,\text{y}\geq0.$

Bag I contains $3$ black and $2$ white balls, Bag II contains $2$ black and $4$ white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

Prove that: $\begin{vmatrix}(b+c)^2&a^2&{bc}\\(c+a)^2&{b}^2&{ca}\\(a+b)^2&{c}^2&{ab}\end{vmatrix}$ $=(a-b)(b-c)(c-b)(a+b+c)({a}^2+{b}^2+{c}^2)$