Integrate: $\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx$

Integrate: ∫ cot⁻¹((1+sin x)/(cos x))dx

Miscellaneous 3 · Q200

\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx

Maharashtra MsbshseTextbookSubjectiveImportance★★★★★

11% · 29/255 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

The standard identity $\frac{1+\sin x}{\cos x}=\tan\left(\frac\pi4+\frac x2\right)$ holds (check at $x=0$ : both sides equal 1). So $\cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)=\cot^{-1}\left[\tan\left(\frac\pi4+\frac x2\right)\right]=\frac\pi2-\left(\frac\pi4+\frac x2\right)=\frac\pi4-\frac x2$ (using $\cot^{-1}(\tan\phi)=\frac\pi2-\phi$ on th …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL2 marks
$\displaystyle\int \sqrt{1+\sin 2x}\cdot dx$
›Reveal solutionSolution

Write $1+\sin2x$ as a perfect square $(\sin x+\cos x)^2$ .

$1+\sin 2x=\sin^2x+\cos^2x+2\sin x\cos x=(\sin x+\cos x)^2$

$\sqrt{1+\sin2x}=\sqrt{(\sin x+\cos x)^2}=|\sin x+\cos x|$

Taking $\sin x+\cos x\ge0$ on the interval considered: …

Start your 14-day free trial to see the full solution →
CBSE 2023Set ANNUAL2 marksMCQ
$\displaystyle\int \cos^3 x\,dx =$
(a) $\dfrac{1}{12}\sin 3x + \dfrac{3}{4}\sin x + c$
(b) $\dfrac{1}{12}\sin 3x + \dfrac{1}{4}\sin x + c$
(c) $\dfrac{1}{12}\sin 3x - \dfrac{3}{4}\sin x + c$
(d) $\dfrac{1}{12}\sin 3x - \dfrac{1}{4}\sin x + c$
›Reveal solutionSolution

Write $\cos^3x=\cos x(1-\sin^2x)$ and integrate, then convert to the $\sin3x$ form.

$\displaystyle\int\cos^3x\,dx=\int\cos x(1-\sin^2x)\,dx=\sin x-\dfrac{\sin^3x}{3}+c$

Using $\sin3x=3\sin x-4\sin^3x \Rightarrow \sin^3x=\dfrac{3\sin x-\sin3x}{4}$ :

…

Start your 14-day free trial to see the full solution →

←Previous questionQ199 — Integrate: ∫ (cos 7x-cos 8x)/(1+2cos 5x)dx →Next questionQ201 — Integrate: ∫ ((1+log x)³)/(x)dx

Q.Integrate: $\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx$

Concept understanding — Reducing Integrals to Standard Form by Algebraic and Trigonometric Simplification

Q.Integrate: ∫cot⁡−1(1+sin⁡xcos⁡x) dx\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx∫cot−1(cosx1+sinx​)dx

🎓Unlock everything free for 14 days

Q.Integrate: $\int \cot^{-1}\left(\frac{1+\sin x}{\cos x}\right)\,dx$

Unlock everything free for 14 days