Integrals of Mixed Powers of Sine and Cosine

Q. 3. Integration of (1/(1+sinx))

Q. If integral $\frac{dx}{(secx+cosecx+tanx+cotx{)}^2}=\frac{x}{a}+\frac{\sqrt{2}cosx+\frac{p}{4}}{b}+\frac{cos2x}{c}+d,$ then a+b+c is equal to

1. -2
2. 2
3. none of these
4. -4

Q. The solution of the differential equation $\left(1+x^2\right)\frac{dy}{dx}+1+y^2=0$, is
(a) tan⁻¹ x − tan⁻¹ y = tan⁻¹ C
(b) tan⁻¹ y − tan⁻¹ x = tan⁻¹ C
(c) tan⁻¹ y ± tan⁻¹ x = tan C
(d) tan⁻¹ y + tan⁻¹ x = tan⁻¹ C

Q. 23.Integration of (1/(1+sinx))

Q.

Find the integrals of the functions.
$\int si{n}^2(2x+5)dx.$

Q.

\(\textstyle \int sin^3(x).cos^5(x)dx\)

Q.

Find x if √3tan5x=1

Q.

Find the integrals of the functions.
$\int si{n}^{4}xdx.$

Q.

Evaluate the definite integrals.

${\int }_{\frac{\pi }{2}}^{\pi }{e}^x\left(\frac{1-sinx}{1-cosx}\right)dx.$

Q.

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\frac{\pi }{2}}\frac{si{n}^{\frac{3}{2}}}{si{n}^{\frac{3}{2}}x+co{s}^{\frac{3}{2}}x}dx.$

Q. Integration of $\frac{1}{1+{\left({\log }_{e}x\right)}^2}$ with respect to log_e x is
(a) $\frac{{\tan }^{-1}\left({\log }_{e}x\right)}{x}+C$

(b) ${\tan }^{-1}\left({\log }_{e}x\right)+C$

(c) $\frac{{\tan }^{-1}x}{x}+C$

(d) none of these

Q. $\int e^x\sec x(1+\tan x)dx$ is equal to
(where $c$ is integration constant)

1. $e^x\sec x+c$
2. $e^x\tan x+c$
3. $e^x(\sec x+\tan x)+c$
4. $e^x(\sec x-\tan x)+c$

Q.

$\int si{n}^{3}x.co{s}^{4}xdx$

1. $\frac{co{s}^7\left(x\right)}{7}-\frac{co{s}^5\left(x\right)}{5}+c$

2. $\frac{co{s}^7\left(x\right)}{5}+\frac{co{s}^5\left(x\right)}{7}+c$

3. $\frac{co{s}^7\left(x\right)}{3}+\frac{co{s}^5\left(x\right)}{5}+c$

4. $\frac{co{s}^7\left(x\right)}{5}+\frac{co{s}^5\left(x\right)}{3}+c$

Q. What is the limit as x tends to a of a rational function f(x)=g(x)÷h(x) where g(a)=h(a)=0?

Q. $\int {\sin }^{3}x{\cos }^{4}xdx$ is equal to.

1. 
   $\frac{1}{7}{\cos }^{7}x+\frac{1}{5}{\cos }^{5}x+C$
2. 
   $\frac{1}{7}{\cos }^{7}x-\frac{1}{5}{\cos }^{5}x+C$
3. 
   $-\frac{1}{7}{\cos }^{7}x+\frac{1}{5}{\cos }^{5}x+C$
4. 
   $-\frac{1}{7}{\cos }^{7}x-\frac{1}{5}{\cos }^{5}x+C$

Q.

$\int si{n}^{3}x.co{s}^{4}xdx$

1. $\frac{co{s}^7\left(x\right)}{7}-\frac{co{s}^5\left(x\right)}{5}+c$

2. $\frac{co{s}^7\left(x\right)}{5}+\frac{co{s}^5\left(x\right)}{7}+c$

3. $\frac{co{s}^7\left(x\right)}{3}+\frac{co{s}^5\left(x\right)}{5}+c$

4. $\frac{co{s}^7\left(x\right)}{5}+\frac{co{s}^5\left(x\right)}{3}+c$

Q. The value of the integral $\int \frac{x^3+x+1}{x^2-1}dx$ is
(where $C$ is an arbitrary constant)

1. $\frac{x^2}{2}+\ln |x+1|+\frac{3}{2}\ln |x-1|+C$
2. $\frac{x^2}{2}-\ln |x+1|+\frac{2}{3}\ln |x-1|+C$
3. $\frac{x^2}{2}+\frac{1}{2}\ln |x+1|+\frac{3}{2}\ln |x-1|+C$
4. $\frac{x^2}{2}+\ln |x-1|+\frac{3}{2}\ln |x+1|+C$

Q. Evaluate }∫ x^2\cdot\operatorname{sin}(x^3)\cdot dx

Q. using properties of definite integral evaluate definite integral of cosx dx/(1+e^x) from -pi/2 to pi/2

Q.
$\text{Find}:\int \sin x\cdot \log \cos xdx.$

Q. $\int \frac{10x-5}{\sqrt{x^2-x+3}dx}$ will be -

1. 5$\sqrt{x^2-x+3}+C$
2. 3$\sqrt{x^2-x+3}+C$
3. 10$\sqrt{x^2-x+3}+C$
4. $\text{None of these}$

Q.

$\int si{n}^{-\frac{11}{3}}\left(x\right).co{s}^{-\frac{1}{3}}\left(x\right)dx$

1. $-\frac{3.co{t}^{\frac{2}{3}}\left(x\right)}{7}-\frac{3.co{t}^{\frac{8}{3}}\left(x\right)}{5}+c$

2. $-\frac{3.co{t}^{\frac{2}{3}}\left(x\right)}{2}-\frac{3.co{t}^{\frac{8}{3}}\left(x\right)}{5}+c$

3. $-\frac{3.co{t}^{\frac{2}{3}}\left(x\right)}{2}-\frac{3.co{t}^{\frac{8}{3}}\left(x\right)}{8}+c$

4. $-\frac{3.co{t}^{\frac{2}{3}}\left(x\right)}{5}-\frac{3.co{t}^{\frac{8}{3}}\left(x\right)}{8}+c$

Q.

How do you solve $\cos \left(2x\right)=-\frac{\sqrt{3}}{2}$ using the double angle identity?

Q. $\int \frac{\mathrm{dx}}{\sqrt[3]{3{\sin }^{11}x\mathrm{cosx}}}\mathrm{is}\mathrm{equal}\mathrm{to}$.

1. $\frac{-3}{8}{\tan }^{-8/3}x-\frac{3}{2}{\tan }^{-2/3}x+C$
2. $\frac{-3}{8}{\tan }^{-8/3}x+\frac{3}{2}{\tan }^{-2/3}x+C$
3. $\frac{3}{8}{\tan }^{-8/3}x+\frac{3}{2}{\tan }^{-2/3}x+C$

Q.

$\int si{n}^{-\frac{11}{3}}\left(x\right).co{s}^{-\frac{1}{3}}\left(x\right)dx$

1. 

2. 

3. 

4. 

Q. The most appropiate substitution for solving the integral $\int {\cos }^{3}x\sqrt{\sin x}dx$ is $t=\sin x$

1. True
2. False

Q. The most appropiate substitution for solving the integral $\int {\cos }^{3}x\sqrt{\sin x}dx$ is $t=\sin x$

1. True
2. False

Q. The value of the integral $\int \frac{2x}{(x^2+1)(x^2+3)}dx$ is
(where $C$ is an arbitrary constant)

Q. Integrate: $si{n}^2(2x+3)$

Q. $\int \frac{x^2}{x^2+6x+12}dx$