Lesson

Integrals

Integration reverses differentiation. Standard integrals, substitution, integration by parts, and partial fractions cover the cases below. Every indefinite integral carries a constant of integration $+C$.

Quiz

Q1

Determine $\displaystyle\int e^{7x}\,\mathrm{d}x$

Show hint

Reverse the chain rule; divide by the exponent's coefficient.

Q2

Determine $\displaystyle\int \frac{1}{3x - 2}\,\mathrm{d}x$

Show hint

The integral of $1/u$ is $\ln|u|$; account for the coefficient.

Q3

Use integration by parts to determine $\displaystyle\int x\sin(x)\,\mathrm{d}x$

Show hint

Let $u = x$ and $\mathrm{d}v = \sin(x)\,\mathrm{d}x$.

Q4

Use partial fractions to determine $\displaystyle\int \frac{8x + 5}{(x-2)(x+5)}\,\mathrm{d}x$

Show hint

Split into $\frac{A}{x-2} + \frac{B}{x+5}$ and solve for $A, B$.