Evaluate the following integrals: ^3x dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\text{x}\cos^3\text{x dx}$
$=\int\text{x}\Big(\frac{3\cos\text{x}+\cos3\text{x}}{4}\Big)\text{dx}$
$\frac{1}{4}\int\text{x}(3\cos\text{x}+\cos3\text{x})\text{dx}$
Using integration by parts,
$\text{I}=\frac{1}{4}\big[\text{x}\int(3\cos\text{x}+\cos3\text{x})\text{dx}-\int(1\int(3\cos\text{x}+\cos3\text{x})\text{dx})\text{dx}\big]$
$=\frac{1}{4}\Big[\text{x}\Big(3\sin\text{x}+\frac{\sin3\text{x}}{3}\Big)-\int\Big(3\sin\text{x}+\frac{\sin3\text{x}}{3}\Big)\text{dx}\Big]$
$=\frac{1}{4}\Big[3\text{x}\sin\text{x}+\frac{\text{x}\sin3\text{x}}{3}+3\cos\text{x}+\frac{\cos3\text{x}}{9}\Big]+\text{C}$
$\text{I}=\frac{3\text{x}\sin\text{x}}{4}+\frac{\text{x}\sin3\text{x}}{12}+\frac{3\cos\text{x}}{4}+\frac{\cos3\text{x}}{36}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the angle between the vectors whose direction cosines are proportional to 2, 3, -6 and 3, -4, 5.

ABCD is a trapezium with AB parallel to DC and DC = 3AB. M is the mid-point of DC,

$\overline{A B}=\bar{p}$ and $\overline{B C}=\bar{q}$. Find in terms of $\bar{p}$ and $\bar{q}$.

(i) $\overline{A M}$

2. $\overline{B D}$

3. $\overline{M B}$

4. $\overline{D A}$

Find the area bounded by the parbola $y = 2 - x^2$ and the strainght line$ y + x = 0.$

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=(\text{x}+1)\text{e}^{-\text{x}}$

Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{k}})+\lambda\hat{\text{i}}+\mu(\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}})$

Evaluate: $\sin\Big\{\cos^{-1}\Big(-\frac{3}{5}\Big)+\cot^{-1}\Big(-\frac{5}{12}\Big)\Big\}$

Solve the following systems of linear equations by cramer's rule:

$x + y + z + w = 2,$

$x - 2y + 2z + 2w = -6,$

$2x + y - 2z + 2w = -5,$

$3x - y + 3z - 3w = -3$

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3- 12x^2 + 36x + 17$

Solve the following differential equation
$\text{e}^{\frac{\text{dy}}{\text{dx}}}=\text{x}+1;\text{y}(0)=3$