Student Mathematics Applications & Interpretation…

What Are the Indefinite and Definite Integrals of x^n; sin x; cos x; and e^x?

The indefinite integral gives a family of antiderivatives and therefore includes a constant, while the definite integral evaluates the change in an antiderivative between two limits. These formulas are HL content in AHL 5.11 Further Integration and may be tested on any GDC-active IB Mathematics: AI HL paper.

Core formulas

Function	Indefinite integral	Definite integral from $a$ to $b$
$x^n$ , where $n \neq -1$	$\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C$	$\displaystyle \int_a^b x^n\,dx=\left[\frac{x^{n+1}}{n+1}\right]_a^b=\frac{b^{n+1}-a^{n+1}}{n+1}$
$x^{-1}=\frac{1}{x}$	$\displaystyle \int \frac{1}{x},dx=\ln	x
$\sin x$	$\displaystyle \int \sin x\,dx=-\cos x+C$	$\int_{a}^{b} \sin x d x =$
$\cos x$	$\displaystyle \int \cos x\,dx=\sin x+C$	$\int_{a}^{b} \cos x d x = [\sin x_{}^{}$
$e^x$	$\displaystyle \int e^x\,dx=e^x+C$

The constant of integration $C$ is needed because differentiating any constant gives zero. A definite integral does not include $C$ , since it cancels when the limits are substituted.

By the fundamental theorem of calculus, if $F'(x)=f(x)$ , then

\int_a^b f(x)\,dx=F(b)-F(a).

To integrate a sum, integrate each term separately and retain any numerical coefficient. The antiderivative can then be checked by differentiation: differentiating your result should return the original integrand. This reverse check is especially useful for signs. The derivative of negative cosine is positive sine, whereas the derivative of sine is cosine. In trigonometric integrals, angles must be measured in radians.

A definite integral is signed area: regions above the horizontal axis contribute positively, while regions below it contribute negatively. Therefore, a definite integral is not always the geometric area. If a graph crosses the horizontal axis and the question asks for total area, split the integral at each root and make every section positive, or integrate the absolute value where appropriate. Also check that the definite integral is proper: the reciprocal rule cannot be applied across zero because the integrand is undefined there.

In AI HL, technology may evaluate the integral, but correct mathematical setup remains essential method evidence.

For example,

\int_0^{\frac{\pi}{2}}(2x^3+\cos x)\,dx =\left[\frac{x^4}{2}+\sin x\right]_0^{\frac{\pi}{2}} =\frac{\pi^4}{32}+1.

Exam technique

Write the antiderivative before using your GDC, substitute the upper limit before subtracting the lower limit, and use radians for trigonometric calculus. A common misconception is applying the power rule to $x^{-1}$ ; this case gives $\ln|x|$ , not division by zero.

What Are the Indefinite and Definite Integrals of x^n; sin x; cos x; and e^x?

Core formulas

Function	Indefinite integral	Definite integral from $a$ to $b$
$x^n$ , where $n \neq -1$	$\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C$	$\displaystyle \int_a^b x^n\,dx=\left[\frac{x^{n+1}}{n+1}\right]_a^b=\frac{b^{n+1}-a^{n+1}}{n+1}$
$x^{-1}=\frac{1}{x}$	$\displaystyle \int \frac{1}{x},dx=\ln	x
$\sin x$	$\displaystyle \int \sin x\,dx=-\cos x+C$	$\int_{a}^{b} \sin x d x =$
$\cos x$	$\displaystyle \int \cos x\,dx=\sin x+C$	$\int_{a}^{b} \cos x d x = [\sin x_{}^{}$
$e^x$	$\displaystyle \int e^x\,dx=e^x+C$

The constant of integration $C$ is needed because differentiating any constant gives zero. A definite integral does not include $C$ , since it cancels when the limits are substituted.

By the fundamental theorem of calculus, if $F'(x)=f(x)$ , then

\int_a^b f(x)\,dx=F(b)-F(a).

In AI HL, technology may evaluate the integral, but correct mathematical setup remains essential method evidence.

For example,

\int_0^{\frac{\pi}{2}}(2x^3+\cos x)\,dx =\left[\frac{x^4}{2}+\sin x\right]_0^{\frac{\pi}{2}} =\frac{\pi^4}{32}+1.

What Are the Indefinite and Definite Integrals of x^n; sin x; cos x; and e^x?

Core formulas

Exam technique

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What Are the Indefinite and Definite Integrals of x^n; sin x; cos x; and e^x?

Core formulas

Exam technique