Evaluate the following integrals: ^3x^2 ^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$

✓

Answer

$\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$
Let $\text{x}^2=\text{t}$
$2\text{x}\text{dx}=\text{dt}$
$\text{x}\text{ dx}=\frac{\text{dt}}{2}$
Now, $\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$
$=\frac{1}{2}\int\cos^3\text{t}\cdot\sin\text{t}\text{ dt}$
Again let $\cos\text{t}=\text{p}$
$-\sin\text{t}\text{ dt}=\text{dp}$
$\sin\text{t}\text{ dt}=-\text{dp}$
So, $\frac{1}{2}\int\cos^3\text{t}\cdot\sin\text{t}\text{ dt}$
$=-\frac{1}{2}\text{p}^3\text{ dp}$
$=-\frac{1}{2}\Big(\frac{\text{p}^4}{4}\Big)+\text{C}$
$=-\frac{\text{p}^4}{8}+\text{C}$
$=-\frac{\cos^4\text{t}}{8}+\text{C}$
$=-\frac{\cos^4\text{x}^2}{8}+\text{C}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that if $A=\left[\begin{array}{ll} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right], \text { then } A^{n}=\left[\begin{array}{cc} \cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta \end{array}\right]$

A kite is $120m$ high and $130m$ of string is out. If the kite is moving away horizontally at the rate of $52m/ \sec,$ find the rate at which the string is being paid out.

Evaluate the following integrals:
$\int^\limits{\text{a}}_{-\text{a}}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}\text{ dx}$

Find whether the function is differentiable at x = 1 and x = 2
$\text{f(x)}=\begin{cases}\text{x} & \text{x}\leq1\\2-\text{x} & 1\leq\text{x}\leq2\\-2+3\text{x}&\text{x}>2\end{cases}$

At what points on the curve $y = 2x^2 - x + 1$ is the tangent parallel to the line $y = 3x + 4?$

Find the coordinates of the foot of the perependicular drawn from the origin to the plane 2x - 3y + 4z - 6 = 0.

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix}$

Find the area of the bounded by the curve xy - 3x - 2y = 0, x-axis and the lins x = 3, x = 4.

If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most $24$. It takes $1$ hour to make a ring and $30$ minutes to make a chain. The maximum number of hours available per day is $16$. If the profit on a ring is $Rs. 300$ and that on a chain is $Rs. 190,$ find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an $\text{LPP}$ and solve it graphically.