Mathematics

# Solve the integral:-$\int {{{\cos }^3}x\, {{\sin }^5}xdx = ?}$

##### SOLUTION
$\displaystyle\int{{\cos}^{3}{x}{\sin}^{5}{x}dx}$

$=\displaystyle\int{\cos{x}{\cos}^{2}{x}{\sin}^{5}{x}dx}$

$=\displaystyle\int{\cos{x}\left(1-{\sin}^{2}{x}\right){\sin}^{5}{x}dx}$

$=\displaystyle\int{\left({\sin}^{5}{x}-{\sin}^{7}{x}\right)\cos{x}dx}$

Let $t=\sin{x}\Rightarrow\,dt=\cos{x}dx$

$=\displaystyle\int{\left({t}^{5}-{t}^{7}\right)dt}$

$=\dfrac{{t}^{6}}{6}-\dfrac{{t}^{8}}{8}+c$

$=\dfrac{{\sin}^{6}{x}}{6}-\dfrac{{\sin}^{8}{x}}{8}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int\frac{1+x^{5}}{1+x}dx$ is equal to
• A. $1-x+x^{2}-x^{3}+x^{4}+c$
• B. $(1+x)^{5}+c$
• C. $(1-x)^{5}+c$
• D. $x-\displaystyle \frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}+c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Solve$\displaystyle \int_{0}^{\pi /2}\left ( 2\log \sin x-\log \sin 2x \right )dx$
• A. $\dfrac {\pi}{2}\log 2$
• B. $\dfrac {\pi}{4}\log 2$
• C. None
• D. $-\dfrac {\pi}{2}\log 2$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Medium
##### ASSERTION

$\displaystyle \int_{0}^{\pi /2}x\cot x\:dx= \displaystyle \frac{\pi }{2}\log 2$

##### REASON

$\displaystyle \int_{0}^{\pi /2}\log \sin x\:dx=- \displaystyle \frac{\pi }{2}\log 2$

• A. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• B. Assertion is true but Reason is false
• C. Assertion is false but Reason is true
• D. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
integrate :
$\int {{{\tan }^{ - 1}}xdx} .$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020