Standard Integrals in Calculus Explained — JAMB Mathematics 

Standard integrals are a set of well-known, memorized integration results for common functions – such as powers, trigonometric functions, and exponentials, that let you integrate quickly without re-deriving the rule each time. 

This topic builds directly on basic integration and is regularly tested in JAMB Mathematics, usually through direct application of one or more standard results. This lesson walks through the most important standard integrals with worked examples.

Quick takeaways

• Standard integrals are memorized results for common function types, so you don’t have to derive them from first principles every time.
• The power rule (∫xⁿ dx = xⁿ⁺¹/(n+1) + C) is the most frequently tested standard integral at JAMB level.
• Trigonometric functions have their own standard results, such as ∫cos x dx = sin x + C and ∫sin x dx = −cos x + C.
• The integral of eˣ is simply eˣ + C, one of the easiest standard integrals to remember.
• JAMB questions usually test one standard integral directly, or ask you to integrate a simple combination of terms using more than one rule.

What are standard integrals?

Standard integrals are a set of established integration results for commonly occurring functions, memorized so they can be applied directly rather than derived from scratch each time. Just as multiplication tables save you from repeatedly adding numbers, standard integrals save you from repeatedly working through the logic of integration for functions you’ll encounter again and again.

What is the standard integral for powers of x?

The most commonly tested standard integral at JAMB level is the power rule:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1)

This applies to any polynomial term, so integrating an expression like 3x² simply means applying this rule to x² and keeping the constant 3 outside the integral.

What are the standard integrals for trigonometric functions?

A few trigonometric standard integrals appear regularly:

• ∫cos x dx = sin x + C
• ∫sin x dx = −cos x + C
• ∫sec²x dx = tan x + C

Notice the sign change when integrating sin x – this is one of the most common places students lose marks, so it’s worth memorizing carefully rather than guessing.

What is the standard integral for exponential functions?

The exponential function eˣ has a particularly simple standard integral:

∫eˣ dx = eˣ + C

This is because eˣ is unique in that its derivative and its integral are both itself, making it one of the easiest standard integrals to apply once you recognize the function.

How do you combine standard integrals in one question?

Many JAMB questions ask you to integrate an expression made up of multiple terms, such as 3x² + cos x. In these cases, you integrate each term separately using its relevant standard integral, then combine the results with a single constant of integration:

∫(3x² + cos x) dx = x³ + sin x + C

The key is recognizing which standard integral applies to each individual term before combining them.

Common mistakes students make with standard integrals

• Forgetting the sign change when integrating sin x (∫sin x dx = −cos x + C, not +cos x)
• Applying the power rule incorrectly to non-polynomial terms like trigonometric or exponential functions
• Adding a separate constant of integration for each term instead of one combined constant at the end
• Mixing up standard integrals with standard derivatives, especially for trigonometric functions

Frequently asked questions

Q: What is the standard integral of xⁿ? 

The standard integral of xⁿ is xⁿ⁺¹/(n+1) + C, provided the exponent n is not equal to −1.

Q: What is the integral of sin x? 

The integral of sin x is −cos x + C — note the negative sign, which is a common point of confusion.

Q: What is the integral of eˣ? 

The integral of eˣ is simply eˣ + C, since the exponential function eˣ is its own derivative and its own integral.

Q: How do you integrate an expression with multiple terms? 

Integrate each term separately using its relevant standard integral, then combine the results under a single constant of integration.

New to integration? Start with our step-by-step definite integral lesson →

This is one topic from our full JAMB Mathematics course — see everything covered →