Evaluate int0fracpi4 fracdx1-sin x, Evaluate int x2 sin x , dx. Solve | Basic Mathematics 2080

Basic Mathematics208010 marksNumerical: integration and differential equationIntegration by parts

Evaluate $\int_{0}^{\frac{\pi}{4}} \frac{dx}{1-\sin x}$ , Evaluate $\int x^2 \sin x \, dx$ . Solve the differential equation $\frac{dy}{dx} - \frac{3y}{x} = x$ , $x > 0$ . [5+5]

Evaluate $\int_{0}^{\frac{\pi}{4}} \frac{dx}{1-\sin x}$, Evaluate $\int x^2 \sin x , dx$. Solve the differential equation $\frac{dy}{dx} - \frac{3y}{x} = x$, $x > 0$. [5+5]

Given data: Integrand $\frac{1}{1-\sin x}$ , limits $0$ to $\pi/4$ . Multiply numerator and denominator by the conjugate $(1+\sin x)$ : $\int0^{\pi/4} \frac{1+\sin x}{(1-\sin x)(1+\sin x)}\,dx = \int0^{\pi/4} \frac{1+\sin x}{1-\sin^2 x}\,dx = \int0^{\pi/4} \frac{1+\sin x}{\cos^2 x}\,dx$ Split: $$=...

Given data: Integrand $\frac{1}{1-\sin x}$, limits $0$ to $\pi/4$. Multiply numerator and denominator by the conjugate $(1+\sin x)$: $$\int0^{\pi/4} \frac{1+\sin x}{(1-\sin x)(1+\sin x)},dx = \int0^{\pi/4} \frac{1+\sin x}{1-\sin^2 x},dx = \int0^{\pi/4} \frac{1+\sin x}{\cos^2 x},dx$$ Split: $$=...

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Evaluate int0fracpi4 fracdx1-sin x, Evaluate int x2 sin x , dx. Solve the differential equation fracdydx - frac3yx = x, x > 0. [5+5]