Question
Evaluate the following integrals:
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
✓
Answer
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{\cos^2\text{x}\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$\text{Let }\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\cos\text{x dx}=\text{dt}$
$\text{Now,}\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\text{t}^2)}{\sqrt{\text{t}}}\text{dt}$
$=\int\Big(\frac{1}{\sqrt{\text{t}}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\int\Big(\text{t}^{-\frac{1}{2}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\Bigg[\frac{\text{t}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}-\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]+\text{C}$
$=2\sqrt{\text{t}}-\frac{2}{5}\text{t}^\frac{5}{2}+\text{C}$
$=2\sqrt{\sin\text{x}}-\frac{2}{5}\sin^\frac{5}{2}\text{x}+\text{C}$
$=\int\frac{\cos^2\text{x}\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$\text{Let }\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\cos\text{x dx}=\text{dt}$
$\text{Now,}\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\text{t}^2)}{\sqrt{\text{t}}}\text{dt}$
$=\int\Big(\frac{1}{\sqrt{\text{t}}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\int\Big(\text{t}^{-\frac{1}{2}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\Bigg[\frac{\text{t}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}-\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]+\text{C}$
$=2\sqrt{\text{t}}-\frac{2}{5}\text{t}^\frac{5}{2}+\text{C}$
$=2\sqrt{\sin\text{x}}-\frac{2}{5}\sin^\frac{5}{2}\text{x}+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Show that the following system of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13
2x + 5y = 7
6x + 15y = 13
There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:
Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-2\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}$
→$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-2\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}$
If $\sin(\text{xy})+\frac{\text{y}}{\text{x}}=\text{x}^2-\text{y}^2,$ find $\frac{\text{dy}}{\text{dx}}$
→Maximize Z = 9x + 3y
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
→Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
In a bank principal increases at the rate of r% par year. Find the value of r if ₹100 double it self in 10 years (loge 2 = 0.6931).
If $\text{x}=\text{a}\cos\theta,\text{y}=\text{b}\sin\theta$ Show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$
→Differentiate $\sin^{-1}\Big(2\text{ax}\sqrt{1-\text{a}^2\text{x}^2}\Big)$ with respect to $\sqrt{1-\text{a}^2\text{x}^2},$ if $-\frac{1}{\sqrt{2}}<\text{ax}<\frac{1}{\sqrt{2}}$.
→$\text{If} \overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{d} and \overrightarrow{a} \times\overrightarrow{c} = \overrightarrow{b} \times\overrightarrow{d},$ show that $\overrightarrow{a} - \overrightarrow{d}$ is parallel to $\overrightarrow{b}-\overrightarrow{c},$ where $\overrightarrow{a} \neq\overrightarrow{d} and \overrightarrow{b}\neq\overrightarrow{c}.$
→Evaluate the following integrals:
$\int\limits^{\text{a}}_0\frac{1}{\text{x}+\sqrt{\text{a}^2-\text{x}^2}}\text{ dx}$
→$\int\limits^{\text{a}}_0\frac{1}{\text{x}+\sqrt{\text{a}^2-\text{x}^2}}\text{ dx}$