Integrate the functions: `cos x /(sqrt(1+sinx))`

Let `I = int (cos x)/sqrt(1 + sin x)` dx Put 1 + sin x = t cos x dx = dt &there4; `I = int dt/sqrt(1 + t) = (1 + t)^(1/2 +1)/(1/2 + 1) + C` `= 2 (1 + t)^(1/2) + C` `= 2 sqrt(1 + sin x + C)`

Integrate the functions: cosx1+sinx - Mathematics

प्रश्न

Integrate the functions:

$\frac{\cos x}{\sqrt{1 + \sin x}}$

बेरीज

उत्तर

Let $I = \int \frac{\cos x}{\sqrt{1 + \sin x}}$ dx

Put 1 + sin x = t

cos x dx = dt

∴ $I = \int \frac{d t}{\sqrt{1 + t}} = \frac{{(1 + t)}^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C$

$= 2 {(1 + t)}^{\frac{1}{2}} + C$

$= 2 \sqrt{1 + \sin x + C}$

shaalaa.com

Methods of Integration: Integration by Substitution

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 28 Q 27 Q 29

पाठ 7: Integrals - Exercise 7.2 [पृष्ठ ३०५]

APPEARS IN

एनसीईआरटी Mathematics [English] Class 12

पाठ 7 Integrals
Exercise 7.2 | Q 28 | पृष्ठ ३०५

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

view
व्हिडिओ ट्यूटोरियल For All Subjects
Methods of Integration: Integration by Substitution

video tutorial00:35:03
Methods of Integration: Integration by Substitution

video tutorial01:32:47
Methods of Integration: Integration by Substitution

video tutorial02:09:40

संबंधित प्रश्‍न

Integrate the functions:

$\sqrt{a x + b}$

Integrate the functions:

$x \sqrt{x + 2}$

Integrate the functions:

$\frac{1}{x {(\log x)}^{m}}, x > 0, m \neq 1$

Integrate the functions:

$\frac{\sin^{- 1} x}{\sqrt{1 - x^{2}}}$

Evaluate: $\int_{0}^{3} f (x) d x$ where f(x) = ${\begin{cases} \cos 2 x & 0 \leq x \leq \frac{π}{2} \\ 3 & \frac{π}{2} \leq x \leq 3 \end{cases}$

\int \sqrt{2 x^{2} + 3 x + 4} dx

Write a value of $\int (e^{x \log_{e} a} + e^{a \log_{e} x}) d x$ .

Write a value of

\int \frac{\cos x}{\sin x \log \sin x} d x

Write a value of $\int \frac{\sin 2 x}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} dx$

Write a value of

\int \frac{a^{x}}{3 + a^{x}} dx

Write a value of $\int e^{a x} \sin b x d x$

Write a value of $\int e^{a x} {a f (x) + f^{'} (x)} d x$ .

$\int \frac{\sin x + 2 \cos x}{2 \sin x + \cos x} dx$

Evaluate the following integrals : $\int \frac{\sin 2 x}{\cos x} d x$

Evaluate the following integrals : $\int \frac{5 x + 2}{3 x - 4} . d x$

Evaluate the following integrals:

$\int \frac{2}{\sqrt{x} - \sqrt{x + 3}} . d x$

Integrate the following functions w.r.t. x : $\frac{\sin x \cos^{3} x}{1 + \cos^{2} x}$

Evaluate the following : $\int \frac{1}{\sqrt{3 x^{2} + 5 x + 7}} . d x$

Evaluate $\int {(3 x^{2} - 5)}^{2}$ dx

Evaluate the following.

$\int {(e^{x} + e^{- x})}^{2} (e^{x} - e^{- x})$ dx

Evaluate the following.

$\int \frac{3 e^{x} + 4}{2 e^{x} - 8}$ dx

Evaluate the following.

$\int \frac{1}{4 x^{2} - 1}$ dx

Evaluate the following.

$\int \frac{1}{x^{2} + 4 x - 5}$ dx

Evaluate the following.

$\int \frac{1}{4 x^{2} - 20 x + 17}$ dx

Evaluate the following.

$\int \frac{1}{7 + 6 x - x^{2}}$ dx

$\int \frac{\log x}{x} d x$

The general solution of the differential equation $(1 + \frac{y}{x}) + \frac{d y}{d x}$ = 0 is ______.

$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x$ is equal to ______. (where C is a constant of integration).

$\int (\log (\log x) + \frac{1}{{(\log x)}^{2}}) d x$ = ______.

If $\int \frac{\sin x}{\sin^{3} x + \cos^{3} x} d x = α \log_{e} | 1 + \tan x | + β \log_{e} | 1 - \tan x + \tan^{2} x | + γ \tan^{- 1} (\frac{2 \tan x - 1}{\sqrt{3}}) + C$ , when C is constant of integration, then the value of 18(α + β + γ²) is ______.

Evaluate $\int_{- a}^{a} f (x) d x$ , where f(x) = $\frac{9^{x}}{1 + 9^{x}}$ .

Evaluate.

$\int \frac{5 x^{2} - 6 x + 3}{2 x - 3} d x$

Evaluate:

$\int (\sqrt{\tan x} + \sqrt{\cot x}) d x$

If f ′(x) = 4x³ − 3x² + 2x + k, f(0) = 1 and f(1) = 4, find f(x)

$\int {cosec}^{4} x d x$ = ______.

Evaluate the following:

$\int \frac{1}{x^{2} + 4 x - 5} d x$

Evaluate $\int (1 + x + \frac{x^{2}}{2!}) d x$

Evaluate $\int \frac{1}{x (x - 1)} d x$

Integrate the functions: cosx1+sinx - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्‍न

Select a course

Integrate the functions: cosx1+sinx - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्‍न

Select a course

उत्तर