CBSE 2024 · Region 3 · Set 1 · Q23 · 2 marks
Evaluate : \[\int_{0}^{\pi / 2} \sin 2 x \cos 3 x d x \]Given $\displaystyle \frac{d}{d x} F(x)=\frac{1}{\sqrt{2 x-x^{2}}}$ and $\displaystyle F(1)=0$, find $\displaystyle F(x)$.
Evaluate : \[\int_{0}^{\pi / 2} \sin 2 x \cos 3 x d x \]
Given $\displaystyle \frac{d}{d x} F(x)=\frac{1}{\sqrt{2 x-x^{2}}}$ and $\displaystyle F(1)=0$, find $\displaystyle F(x)$.
Marking-scheme solution
$$\begin{aligned}
& I=\int_{0}^{\frac{\pi}{2}} \sin 2 x \cos 3 x d x \\
& =\frac{1}{2} \int_{0}^{\frac{\pi}{2}}(\sin 5 x-\sin x) d x \\
& =\frac{1}{2}\left[-\frac{1}{5} \cos 5 x+\cos x\right]_{0}^{\frac{\pi}{2}} \\
& =-\frac{2}{5}
\end{aligned}
\begin{aligned}
& F(x)=\int \frac{1}{\sqrt{2 x-x^{2}}} d x \\
& =\int \frac{1}{\sqrt{1-(x-1)^{2}}} d x \\
& =\sin ^{-1}(x-1)+c \\
& \text { when } x=1, y=0 \text { gives } c=0 \\
& \therefore F(x)=\sin ^{-1}(x-1)
\end{aligned}
$$
IntegralsEvaluation of Definite Integrals by SubstitutionApplyvery_short_answermedium
More from Integrals
- ∫ (x+5)/((x+6)^2) e^x dx is equal to:2025 · asked 3×
- Evaluate ∫ 1/((e^x+e^-x)(e^x-e^-x)) d x log √22023 · asked 3×
- If ∫ (3 a x)/( b^2+c^2 x^2) d x=A log b^2+c^2 x^2 +K, then the value of A is2026 · asked 3×
- Find: Find: ∫ x^2 log (x^2+1) d x2023 · asked 3×
- If ∫ (2^1/x)/(x^2) d x=k · 2^1/x+C, then k is equal to2025 · asked 3×
- Find: ∫ (x^2+1)/((x-1)^2(x+3)) d x OR Evaluate: ∫ 0^π / 2 (x)/( sin x+ cos x) d x2025 · asked 3×
- Find: ∫ dx√4 x-x^22022 · asked 3×
- Find: ∫ (2 x)/((x^2+3)(x^2-5)) d x OR Evaluate: ∫ 1^4( x-2 + x-4 ) d x2025 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.