Evaluate the following integrals: ^ 2 _0 ^5x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\cos^5\text{x dx}$

✓

Answer

$\int^\limits{\frac{\pi}{2}}_0\cos^5\text{x dx}=\int^\limits{\frac{\pi}{2}}_0\big(1-\sin^2\text{x}\big)^2\cos\text{x dx}$
Let $\sin\text{x}=\text{t}$
Differentiating w.r.t. x, we get
$\cos\text{x dx}=\text{dt}$
When $\text{x}=0\Rightarrow\text{t}=0$
$\text{x}=\frac{\pi}{2}\Rightarrow\text{t}=1$
$=\int^\limits{\frac{\pi}{2}}_0\big(1-\sin^2\text{x}\big)^2\cos\text{x dx}$
$=\int^\limits{1}_0\big(1-\text{t}^2\big)^2\text{ dt}$
$=\int^\limits{1}_0\big(1-2\text{t}^2+\text{t}^4\big)\text{dt}$
$=\Big[\text{t}-\frac{2}{3}\text{t}^3+\frac{\text{t}^5}{5}\Big]^1_0$
$=1-\frac{2}{3}+\frac{1}{5}$
$=\frac{8}{15}$

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 "4 Marks" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec{\text{a}},\vec{\text{ b}}$ and $\vec{\text{c}}$ determine the vertices of a triangle, show that $\frac{1}{2}[\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}+\vec{\text{a}}\times\vec{\text{b}}]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\vec{\text{a}},\vec{\text{ b}}$ and $\vec{\text{c}}$ are collinear. Also, find the unit vector normal to the plane of the triangle.

Show that the lines $\frac{\text{x}-1}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-1}{5}$ and $\frac{\text{x}+2}{4}=\frac{\text{y}-1}{3}=\frac{\text{z}+1}{-2}$ do not intersect.

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab} \end{vmatrix}$

Evaluate the following definite integrals:
$\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$

Prove that the curves xy = 4 and x² + y² = 8 touch each other.

Using properties of definite integrals, prove the following:

$\int\limits_0^{\pi} \frac{x \tan x}{\sec x\text{ }cosec\text{ x}} dx = \frac{\pi^{2}}{4}$

Consider the binary operation * and o defined by the following tables on set S = {a, b, c, d}.

*	a	b	c	d
a	a	b	c	d
b	b	a	d	c
c	c	d	a	b
d	d	c	b	a

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs. 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Show that the function f defined as follows,
$\text{f(x)}=\begin{cases}3\text{x}-2, & 0<\text{x}\leq1\\2\text{x}^2-\text{x,} & 1<\text{x}\leq2\\5\text{x}-4,&\text{x}>2\end{cases}$
is countinuous at x = 2, but not differentiable there at x = 2.

Find the particular solution of the differential equation

$(1-\text{y}^2)(1+\log\text{x})\text{dx}+2\text{xy dy}=0,$ given that $\text{y}=0$ when $\text{x}=1.$