Evaluate the following integrals: ^3 x dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\cos^3\sqrt{\text{x}}\text{dx}$
Let $\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt }$
$=2\int\text{t}\cos^3\text{t dt}$
$=2\int\text{t}\Big(\frac{3\cos\text{t}+\cos3\text{t}}{4}\Big)\text{dt}$
$=\frac{1}{2}\int\text{t}(3\cos\text{t}+\cos3\text{t})\text{dt}$
Using integral\tion by parts,
$\text{I}=\frac{1}{2}\Big[\text{t}\Big(3\sin\text{t}+\frac{1}{3}\sin3\text{t}\Big)+\int\Big(1\times3\sin\text{t}+\frac{\sin3\text{t}}{3}\Big)\text{dt}\Big]$
$=\frac{1}{2}\Big[\text{t}\Big(\frac{9\sin\text{t}+\sin3\text{t}}{3}\Big)+3\cos\text{t}+\frac{\cos3\text{t}}{9}\Big]+\text{C}$
$=\frac{1}{18}\big[27\text{t}\sin\text{t}+3\text{t}\sin3\text{t}+9\cos\text{t}+\cos3\text{t}\big]+\text{C}$
$\text{I}=\frac{1}{18}\big[27\sqrt{\text{x}}\sin\sqrt{\text{x}}+3\sqrt{\text{x}}\sin3\sqrt{\text{x}}+9\cos\sqrt{\text{x}}+\cos3\sqrt{\text{x}}\big]+\text{C}$

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 Indefinite Integrals→Indefinite 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 the function $\text{f}(\text{x})=\cot^{-1}(\sin\text{x}+\cos\text{x})$ is decreasing on $\Big(0,\frac{\pi}{4}\Big)$ and increasing on $\Big(\frac{\pi}{4},\frac{\pi}{2}\Big).$

If $(\cos\text{x})^{\text{y}}=(\tan\text{y})^{\text{x}},$ Prove that $\frac{\text{dy}}{\text{dx}}=\frac{\log\tan\text{y}-\text{y}\tan\text{x}}{\log\cos\text{x}-\text{x}\sec\text{y cosec y}}$

The scalar product of the vector $\hat i + \hat j + \hat k$ with a unit vector along the sum of vectors $2\hat i + 4\hat j - 5\hat k\;$ and $\lambda \hat i + 2\hat j + 3\hat k$ is equal to one. Find the value of$\;\lambda $.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\tan^{-1}\text{x}\text{ on }[0,1]$

Find the maximum and the minimum values, if any, without using derivaives of the following functions: $f(x) = x^3 - 1$ on $R.$

Find the vector equation of the following planes in non-parametric form.
$\vec{\text{r}}=(\lambda-2\mu)\hat{\text{i}}+(3-\mu)\hat{\text{j}}+(2\lambda+\mu)\hat{\text{k}}$

In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y(in milligrams per tablet) are given as below:

Tablets	Iron	Calcium	Vitamin
x	6	3	2
y	2	3	4

The person needs at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligram of vitamins. The price of each tablet of X and Y is Rs. 2 and Rs. 1 respectively. How many tablets of each should the person take inorder to satisfy the above requirement at the minimum cost?

Draw a rough sketch of the region $\{(x, y) : y^2 < 5x, 5x^2 + < 36\}$ and find the area by the region using mwthod of integration.

In each of the form a differential equation representing the given family of curves by eliminating arbitrary constants $a$ and $b.$
$y = ae^{3x} + be^{–2x}$

If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$