Evaluate: ∫ dfracx-2√(x+5)dx – Class 12 Mathematics | ConceptFirst

3.1 · Q19

\int \dfrac{x-2}{\sqrt{x+5}}\,dx

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Let $t=x+5$ , so $x=t-5$ and $x-2=t-7$ .

$\int\dfrac{t-7}{\sqrt t}\,dt=\int(t^{1/2}-7t^{-1/2})dt=\dfrac23t^{3/2}-14t^{1/2}$ . …

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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: …

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

…

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