Integration

Use partial fractions to find $\displaystyle \int \frac{3x - 7}{x^2 - 2x - 15} \, dx.$

Using the substitution $\displaystyle u = \sin \theta$, or otherwise, evaluate

$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 2 \sin^4 \theta \cos \theta \, d\theta.$

Use integration by parts to find $\displaystyle \int x^4 \ln x \, dx$, $\displaystyle x > 0.$

Obtain $\displaystyle \int x^7(\ln x)^2 \,dx.$

On a suitable domain, a curve is defined by the equation $\displaystyle 4x^2 + 9y^2 = 36.$ A section of the curve in the first quadrant, illustrated in the diagram below, is rotated $\displaystyle 360^\circ$ about the $\displaystyle y$-axis.

Calculate the exact value of the volume generated.

(a)Use integration by parts to find the exact value of

$\displaystyle \int_{0}^{1}(x^2 - 2x + 1)e^{4x}dx.$

(b)A solid is formed by rotating the curve with equation $\displaystyle y = 4(x - 1)e^{2x}$ between $\displaystyle x = 0$ and $\displaystyle x = 1$ through $\displaystyle 2\pi$ radians about the $\displaystyle x$-axis.
Find the exact value of the volume of this solid.

Find $\displaystyle \int \frac{x^2}{x^3+10}dx.$

Use the substitution $\displaystyle u = \tan 2x$ to evaluate

$\displaystyle \int_{0}^{\frac{\pi}{8}} \frac{\sqrt{\tan 2x}}{\cos^2 2x} \,dx.$

A solid is formed by rotating part of the curve with equation $\displaystyle y = \frac{1}{\sqrt{1+x^2}}$ about the $\displaystyle x$-axis through $\displaystyle 2\pi$ radians, from $\displaystyle x = 0$ to $\displaystyle x = a.$

The value of the volume of the solid is $\displaystyle \frac{\pi^2}{3}.$

Determine the value of $\displaystyle a.$

Find

$\displaystyle \int\frac{2x+4}{x^{2}+4}\,dx.$