Definite Integral Questions (7)

Evaluate $$\int_{1}^{2} 2x dx$$

$\int_{-\pi/2}^{\pi/2} \sin^4 x \cos^6 x \, dx =$

If $f(x) = \begin{cases} x, & x < 1 \\ \sin x, & x \geq 1 \end{cases}$, find $$\int_{0}^{\pi} f(x) dx$$

$\int_{0}^{1} \frac{\tan^{-1} x}{x} \, dx = \lambda \int_{0}^{\pi/2} \frac{\theta}{\sin \theta} \, d\theta$, then find $\lambda$.

Evaluate $$\int_{0}^{1} \frac{\sin^{-1} x}{\sqrt{1-x^2}} dx$$

If $\frac{d}{dx} f(x) = \frac{e^{\sin x}}{x}, x > 0$ & $\int_{1}^{4} \frac{2e^{\sin x^2}}{x} \, dx = f(k) - f(1)$, then find possible values of $k$.

Evaluate $$\int_{0}^{\ln 2} \frac{e^x}{1+e^x} dx$$