Evaluate the following integrals: ^3x ^5x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sin^3\text{x}\cos^5\text{x}\text{ dx}$

✓

Answer

$\int\sin^3\text{x}\cos^5\text{x}\text{ dx}$
$=\int\sin^2\text{x}\cdot\cos^5\text{x}\cdot\sin\text{x}\text{ dx}$
$=\int(1-\cos^2\text{x})\cdot\cos^5\text{x}\sin\text{x}\text{ dx}$
Let $\cos\text{x}=\text{t}$
$-\sin\text{x}\text{ dx}=\text{dt}$
$\sin\text{x}\text{ dx}=-\text{dt}$
Now, $\int(1-\cos^2\text{x})\cdot\cos^5\text{x}\sin\text{x}\text{ dx}$
$=-\int(1-\text{t}^2)\text{t}^5\text{dt}$
$=-(\text{t}^5-\text{t}^7)\text{dt}$
$=-\int(\text{t}^7-\text{t}^5)\text{dt}$
$=\frac{\text{t}^8}{8}-\frac{\text{t}^6}{6}+\text{C}$
$=\frac{\cos^8\text{x}}{8}-\frac{\cos^6\text{6}\text{x}}{6}+\text{C}$

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 "3 Marks Question" 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

If $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix},$ verify that $A^TA = I_2$.

Find the angle between the following pair of lines:

$\frac{\text{x}-2}{2}=\frac{\text{y}-1}{5}=\frac{\text{z}+3}{-3}\ \text{and}\ \frac{\text{x}+2}{-1}=\frac{\text{y}-4}{8}=\frac{\text{z}-5}{4}$

The cartesian equations of a line are x = ay + b, z = cy + d. Find its direction ratios and reduce it to vector form.

If $x ^{ y }= e ^{ x - y }$, then prove that $\frac{d y}{d x}=\frac{\log x}{(1+\log x)^2}$.

If $\vec{\text{a}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}},$ find $\big(\vec{\text{a}}+2\vec{\text{b}}\big)\times\big(\vec{\text{a}}-2\vec{\text{b}}\big).$

Find mean $u,$variance $\sigma^{2}$for the following probability distribution:

X	0	1	2	3
P(X)	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Solve the equation $\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x, x>0$.

In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.

A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?

What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$