What Are the Indefinite Integrals of x^n; Sin x; Cos x; 1/x; and e^x?

The required indefinite integrals are $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq -1$, $\int \sin x\,dx=-\cos x+C$, $\int \cos x\,dx=\sin x+C$, $\int \frac{1}{x}\,dx=\ln|x|+C$, and $\int e^x\,dx=e^x+C$.

An indefinite integral is a family of functions whose derivative equals the integrand. The constant of integration, $C$, must be included because constants disappear when differentiated. For example, both $x^2$ and $x^2+7$ have derivative $2x$.

For the power rule, increase the exponent by one, then divide by the new exponent. Constants multiplying the integrand remain as constant factors:

$$\int 3x^4\,dx=3\left(\frac{x^5}{5}\right)+C=\frac{3}{5}x^5+C.$$

Differentiate the result to check it: $\frac{d}{dx}\left(\frac{3}{5}x^5+C\right)=3x^4$.

These rules reverse the corresponding differentiation rules. For instance, because differentiating $x^{n+1}$ produces the factor $n+1$, division by $n+1$ is needed when integrating. Similarly, $e^x$ remains unchanged because it is its own derivative. This reverse-differentiation check is especially useful when coefficients or negative exponents are involved.

For example,

$$\int \left(2x^3-4\cos x+5e^x\right)\,dx=\frac{x^4}{2}-4\sin x+5e^x+C.$$

Integration is linear, so each term can be integrated separately. However, only one constant of integration is needed for the complete expression because several arbitrary constants can be combined into a single constant.

The case $n=-1$ is an exception because the power rule would require division by zero. Therefore,

$$\int x^{-1}\,dx=\int \frac{1}{x}\,dx=\ln|x|+C.$$

The modulus is necessary because $1/x$ has antiderivatives on both positive and negative intervals, although no interval may cross $x=0$.

A common misconception is writing $\int \sin x\,dx=\cos x+C$. This loses the negative sign: since $\frac{d}{dx}(\cos x)=-\sin x$, the correct antiderivative is . Another error is adding to a definite integral; it is required only for an indefinite integral.

Exam technique: This is Mathematics AA SL 5.10 and may appear on Paper 1 or Paper 2. Show the antiderivative before applying limits or a boundary condition. Include $C$, preserve $\ln|x|$, and verify the answer by differentiation.