Use a substitution to simplify an integral.

Calculating the area enclosed between a linear function and a quadratic function by integration. The limits (points of intersection) are not given in the question and must be calculated.

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

abc

Calculating the integral of a function of the form $ax^b$ using a table of integrals. 

Calculating the integral of a function of the form $a_1x^{b_1}+a_2x^{b_2}+a_3x^{b_3}$ using a table of integrals. 

Calculating the integral of a function of the form $\frac{a}{bx}$ using a table of integrals.

Integrating an exponential function of the form $ae^{bx}$ using a table of integrals / anti-derivatives.

Calculating the integral of a function of the form $a \sin(bx)$ using a table of integrals.

Calculating the integral of a function of the form $a \cos(bx)$ using a table of integrals.

Calculating the integral of a function of the form $a \sec^2(bx)$ using a table of integrals.

Calculating the integral of a function of the form $\frac{a}{\sqrt{b^2-x^2}}$ using a table of integrals.

Calculating the integral of a function of the form $2ax(ax^2+b)^n$ using integration by substitution.

Calculating the integral of a function of the form $x(ax^2+b)^n$ using integration by substitution.

Calculating the integral of a function of the form $kx e^{ax^2}$ using integration by substitution, where $k=2a$.

Calculating the integral of a function of the form $kx e^{ax^2}$ using integration by substitution, where $k=2na$.

Calculating the integral of a function of the form $n x e^{ax^2}$ using integration by substitution.

Calculating the integral of a function of the form $\frac{nx^{n-1}}{x^n+a}$ using integration by substitution.

Calculating the integral of a function of the form $\frac{\cos(x)}{\sin(x)+a}$ using integration by substitution.

Calculating the integral of a function of the form $\frac{k(2ax+b)}{ax^2+bx+c}$ using integration by substitution.

Calculating the integral of a function of the form $e^x \sin(e^x)$ using integration by substitution.

Calculating the integral of a function of the form $\cos(x) \sin^2(x)$ using integration by substitution.

Calculating the integral of a function of the form $ax \cos(x^2+b)$ using integration by substitution.

Calculating the integral of a function of the form $\frac{\cos(x) \sin(x)}{(\sin(x)+a)^2}$ using integration by substitution.

Calculating the integral of a function of the form $2ax(ax^2+b)^n$ using integration by substitution. $n$ is a positive or negative fraction.

Calculating the integral of a function of the form $\frac{nx^{n-1}}{x^n+a}$ using integration by substitution. $n$ is a positive or negative fraction.

Calculating the integral of a function of the form $\frac{c}{(x+a)(x+b)}$ using partial fractions.

Calculating the integral of a function of the form $\frac{c}{(kx+a)(x+b)}$ using partial fractions.

Calculating the integral of a function of the form $\frac{cx+d}{(x+a)(x+b)}$ using partial fractions.

Calculating the integral of a function of the form $\frac{cx+d}{(x+a)^2}$ using partial fractions.