Antiderivatives and Integrals

Basic Antiderivative Rules

The power rule for derivatives multiplies by the power and then subtracts one from the power. Reversing these operations means that we add one to the power and divide by the new power.

Reverse Power Rule

THEOREM

Reverse Power Rule

\displaystyle \begin{aligned}\int x^n \ dx = {x^{n+1} \over n+1} + C,\quad\text{where }n \ne -1\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int x^6 \ dx

(b)

\displaystyle \int \sqrt[4]{t} \ dt

(c)

\displaystyle \int {1 \over x^{5/3}} \ dx

Antiderivative of Zero

THEOREM

Antiderivative of Zero

\displaystyle \begin{aligned}\int 0 \ dx = C\end{aligned}

Antiderivative of a Constant

THEOREM

Antiderivative of a Constant

\displaystyle \begin{aligned}\int k \ dx = kx + C,\quad\text{where }k\text{ is any constant}\end{aligned}

Find the antiderivatives.

\displaystyle \int \pi \ dx

Multiplicative Constants

THEOREM

Multiplicative Constants

\displaystyle \begin{aligned}\int kf(x) \ dx = k \int f(x) \ dx\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int 4x^7 \ dx

(b)

\displaystyle \int {\pi \over \sqrt{t}} \ dt

Sum / Difference

THEOREM

Sum / Difference

\displaystyle \begin{aligned}\int (f(x) \pm g(x)) \ dx = \int f(x) \ dx \pm \int g(x) \ dx\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int (3x^2 + 5) \ dx

(b)

\displaystyle \int \left({1 \over x^3} - {2 \over x^2}\right) \ dx

QUESTION

Which expression equals

\displaystyle \int(3x^2+5)\,dx

◀ Previous SectionThe Indefinite Integral Next Section ▶Trigonometric and Exponential Integrals

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Antiderivatives and Integrals

Basic Antiderivative Rules

The power rule for derivatives multiplies by the power and then subtracts one from the power. Reversing these operations means that we add one to the power and divide by the new power.

Reverse Power Rule

THEOREM

Reverse Power Rule

\displaystyle \begin{aligned}\int x^n \ dx = {x^{n+1} \over n+1} + C,\quad\text{where }n \ne -1\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int x^6 \ dx

(b)

\displaystyle \int \sqrt[4]{t} \ dt

(c)

\displaystyle \int {1 \over x^{5/3}} \ dx

Antiderivative of Zero

THEOREM

Antiderivative of Zero

\displaystyle \begin{aligned}\int 0 \ dx = C\end{aligned}

Antiderivative of a Constant

THEOREM

Antiderivative of a Constant

\displaystyle \begin{aligned}\int k \ dx = kx + C,\quad\text{where }k\text{ is any constant}\end{aligned}

Find the antiderivatives.

\displaystyle \int \pi \ dx

Multiplicative Constants

THEOREM

Multiplicative Constants

\displaystyle \begin{aligned}\int kf(x) \ dx = k \int f(x) \ dx\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int 4x^7 \ dx

(b)

\displaystyle \int {\pi \over \sqrt{t}} \ dt

Sum / Difference

THEOREM

Sum / Difference

\displaystyle \begin{aligned}\int (f(x) \pm g(x)) \ dx = \int f(x) \ dx \pm \int g(x) \ dx\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int (3x^2 + 5) \ dx

(b)

\displaystyle \int \left({1 \over x^3} - {2 \over x^2}\right) \ dx

QUESTION

Which expression equals

\displaystyle \int(3x^2+5)\,dx

◀ Previous SectionThe Indefinite Integral Next Section ▶Trigonometric and Exponential Integrals