Integrals of Logarithmic Functions

List of integrals involving logarithmic functions \( \int \ln(cx)dx = x\ln(cx) – x \)   \( \int \ln(ax+b)dx = x\ln(ax+b) – x + \frac{b}{a}\ln(ax + b) \)   \( \int (\ln x)^2dx = x(\ln x)^2 – 2x\ln x + 2x \)   \( \int (\ln (cx))^ndx = x(\ln x)^n – n\cdot\int (\ln (cx))^{n-1}dx \)   \( \int \frac{dx}{\ln x} […]

Types of Problems That Cannot Be Solved Without Calculus

Students who go to high school are pretty able to solve basic math problems although calculus is known as difficult field of mathematics, there are no special problems to learn calculus if you go step by step. In mathematics there are so many problems that you can not solve without knowing about calculus. Derivatives Derivative […]

Integrals of Exponential Functions

\( \int e^{cx}dx = \frac{1}{c}e^{cx} \)   \( \int a^{cx}dx = \frac{1}{c\cdot \ln a}a^{cx}, (\text{for } a>0, a\ne1 ) \)   \( \int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1) \)   \( \int x^2 \cdot e^{cx} = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2} + \frac{2}{c^3}\right) \)   \( \int x^n \cdot e^{cx}dx = \frac{1}{c}x^ne^{cx}-\frac{n}{c}\int x^{n-1}e^{cx} dx \)   \( \int \frac{e^{cx}}{x} dx = \ln|x| + \sum\limits_{i=1}^\infty […]

Integrals of Rational Functions

Integrals involving \(ax + b\) \( \int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)}, \quad (\text{for } n \ne 1) \) \( \int \frac{1}{ax+b}dx = \frac{1}{a}\ln|ax+b| \) \( \int x (ax+b)^ndx = \frac{a(n+1)x-b}{a^2(n+1)(n+2)}(ax+b)^{n+1}, \quad (\text{for } n \ne -1, n\ne-2) \) \( \int \frac{x}{ax+b}dx = \frac{x}{2} – \frac{b}{a^2}\ln|ax+b| \) \( \int \frac{x}{(ax+b)^2}dx = \frac{b}{a^2(ax+b)} – \frac{1}{a^2}\ln|ax+b| \) \( \int \frac{x^2}{ax+b} dx = \frac{1}{a^3} […]

Common Integrals

Indefinite Integral Method of substitution \( \int f\left(g(x)\right)\cdot g'(x) dx = \int f(u) du \) Integration by parts \( \int f(x) \cdot g'(x)dx = f(x) \cdot g(x) – \int g(x) \cdot f'(x)dx \) Integrals of Rational and Irrational Functions \( \int x^n dx = \frac{x^{n+1}}{n+1} + C , n \ne 1 \) \( \int \frac{1}{x} dx = \ln|x| […]

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