Mathematics

# $\int x ^ { 2 } \cdot \cos \left( x ^ { 3 } \right) \sqrt { \sin ^ { 7 } \left( x ^ { 3 } \right) } \cdot d x$

##### SOLUTION
$\\Let\>sin(x^3)=t\\then\>\>cosx^3\times\>3x^2\>dx=dt\\or\>x^2cos(x^3)dx=(\frac{dt}{3})\\\therefore\>I=(\frac{1}{3})\int\sqrt{t^7}dt\\=(\frac{1}{3})\int{t^{(\frac{7}{2})}}dt\\=(\frac{1}{3})\times(\frac{t^{(\frac{9}{2})}}{(\frac{9}{2})})+C\\=(\frac{2}{27})\sqrt[9]{sin(x^3)}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int \dfrac {dx}{\sin x. \sin (x + a)}$ is equal to
• A. $cosec \, a \, ln \left |\dfrac {\sin (x + a)}{\sin x}\right | + C$
• B. $cosec \, a \, ln \left |\dfrac {\sin (x + a)}{\sec x}\right | + C$
• C. $cosec \, a \, ln \left |\dfrac {\sec x}{\sec (x + a)}\right | + C$
• D. $cosec \, a \, ln \left |\dfrac {\sin x}{\sin (x + a)}\right | + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac{dt}{t + \sqrt{a^2 - t^2}}$ equal to
• A. $\dfrac{1}{2} \sin^{-1} \left(\dfrac{t}{a} \right) + \log (t + \sqrt{a^2 - t^2}) + k$
• B. $\dfrac{1}{2} \sin^{-1} \left(\dfrac{t}{a} \right) + \log \sqrt{a + \sqrt{a^2 - t^2}} + k$
• C. $\dfrac{1}{2} \sin^{-1} \left(\dfrac{t}{a} \right) + \log (a + \sqrt{a^2 - t^2}) + k$
• D. $\dfrac{1}{2} \sin^{-1} \left(\dfrac{t}{a} \right) + \log \sqrt{t + \sqrt{a^2 - t^2}} + k$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\displaystyle I = \int \frac {\cos x \: dx}{\sqrt {a + b \cot^2 x}} (a > b > 0)$, then I equals
• A. $\displaystyle \frac {1}{a - b} \sqrt {a + b \cot^2 x} + C$
• B. $\displaystyle \frac {1}{a - b} \left ( \sqrt {a + b \cot^2 x} + x \right ) + C$
• C. $\displaystyle \frac {1}{a - b} \left ( \sqrt {a + b \cot^2 x} - x \right ) + C$
• D. $\displaystyle \frac {1}{a - b} \sqrt {a \sin^2 x + b \cos^2 x} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int \dfrac { e ^ { x } } { x } \left( x \cdot ( \log x ) ^ { 2 } + 2 \log x \right) d x$

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 $\displaystyle f(x)>0 \forall x \in R$