10 Antiderivative Practice Problems with Full Solutions

💡 Reminder Before You Start: Every indefinite integral answer must include + C — the constant of integration. It is a required part of every complete answer throughout all ten problems below. If your answer is missing it, the answer is technically incomplete regardless of how correct the rest of it is.

Full Solution:

Rule to use: Power rule — ∫ xⁿ dx = x^(n+1) / (n+1) + C.

Step 1: Identify the exponent: n = 5.

Step 2: Add 1 to the exponent: 5 + 1 = 6. So x⁵ becomes x⁶.

Step 3: Divide by the new exponent: x⁶ / 6.

Step 4: Add + C.

✅ Answer: x⁶/6 + C

Check: Differentiate x⁶/6 + C → (6x⁵)/6 = x⁵. ✓

Full Solution:

Rule to use: Power rule combined with the constant multiple rule.

Step 1: Identify the exponent: n = 3. Coefficient: 8.

Step 2: Add 1 to the exponent: 3 + 1 = 4. So x³ becomes x⁴.

Step 3: Divide by the new exponent: x⁴ / 4.

Step 4: Multiply by the coefficient: 8 × (x⁴/4) = 2x⁴.

Step 5: Add + C.

✅ Answer: 2x⁴ + C

Check: Differentiate 2x⁴ + C → 8x³. ✓

Full Solution:

Rule to use: Sum and difference rule — integrate each term separately, then combine.

Step 1: Split into three terms: 6x², 4x, and 7.

Step 2: Antiderivative of 6x².
Add 1 to exponent: 2 + 1 = 3. Divide: x³/3. Multiply by 6: 6 × (x³/3) = 2x³.

Step 3: Antiderivative of 4x.
Add 1 to exponent: 1 + 1 = 2. Divide: x²/2. Multiply by 4: 4 × (x²/2) = 2x².

Step 4: Antiderivative of 7.
Constant rule: 7x.

Step 5: Combine all terms, apply subtraction, and add + C.

✅ Answer: 2x³ − 2x² + 7x + C

Check: Differentiate → 6x² − 4x + 7. ✓

Full Solution:

Rule to use: Power rule — works for all exponents except n = −1.

Step 1: Identify the exponent: n = −4.

Step 2: Add 1 to the exponent: −4 + 1 = −3. So x⁻⁴ becomes x⁻³.

Step 3: Divide by the new exponent: x⁻³ / (−3) = −(1/3)x⁻³.

Step 4: Add + C.

Step 5 (optional simplification): Rewrite x⁻³ as 1/x³: −1/(3x³).

✅ Answer: −(1/3)x⁻³ + C  or equivalently  −1/(3x³) + C

Check: Differentiate −(1/3)x⁻³ + C → −(1/3) · (−3x⁻⁴) = x⁻⁴. ✓

Full Solution:

Rule to use: Power rule — first rewrite √x as a fractional exponent.

Step 1: Rewrite √x as x^(1/2).

Step 2: Identify the exponent: n = 1/2.

Step 3: Add 1 to the exponent: 1/2 + 1 = 1/2 + 2/2 = 3/2. So x^(1/2) becomes x^(3/2).

Step 4: Divide by the new exponent: x^(3/2) ÷ (3/2) = x^(3/2) × (2/3) = (2/3)x^(3/2).

Step 5: Add + C.

✅ Answer: (2/3)x^(3/2) + C

Check: Differentiate → (2/3) · (3/2)x^(1/2) = x^(1/2) = √x. ✓

Full Solution:

Rule to use: Natural logarithm rule — the power rule breaks down at n = −1.

Step 1: Recognize that 1/x = x⁻¹, which means n = −1. The power rule cannot be used.

Step 2: Apply the natural log rule: ∫ (1/x) dx = ln|x|.

Step 3: Add + C.

✅ Answer: ln|x| + C

Check: Differentiate ln|x| + C → 1/x. ✓

Note: The absolute value bars in ln|x| are essential — they extend the validity of the result to negative values of x as well as positive ones.

Full Solution:

Rule to use: Trig antiderivative rules combined with the sum/difference and constant multiple rules.

Step 1: Split into two separate terms: 3cos(x) and 2sin(x).

Step 2: Antiderivative of 3cos(x).
∫ cos(x) dx = sin(x). Multiply by 3: 3sin(x).

Step 3: Antiderivative of 2sin(x).
∫ sin(x) dx = −cos(x). Multiply by 2: −2cos(x).

Step 4: Combine both results, apply subtraction, and add + C.
3sin(x) − (−2cos(x)) = 3sin(x) + 2cos(x).

✅ Answer: 3sin(x) + 2cos(x) + C

Check: Differentiate → 3cos(x) + 2(−sin(x)) = 3cos(x) − 2sin(x). ✓

Full Solution:

Rule to use: Exponential rule for e^(ax) combined with the constant multiple rule.

Step 1: Identify the inner coefficient: a = 2. Outer coefficient: 5.

Step 2: Apply the exponential rule: ∫ e^(ax) dx = (1/a) · e^(ax).

Step 3: Substitute a = 2: (1/2) · e^(2x).

Step 4: Multiply by the outer coefficient 5: 5 × (1/2) · e^(2x) = (5/2)e^(2x).

Step 5: Add + C.

✅ Answer: (5/2)e^(2x) + C

Check: Differentiate using the chain rule → (5/2) · e^(2x) · 2 = 5e^(2x). ✓

Full Solution:

Rule to use: U-substitution — look for a function and something close to its derivative in the same integral.

Step 1: Spot the pattern. The inner function is x² + 1. Its derivative is 2x. The integral contains 6x, which is 3 times 2x — so the derivative is present up to a constant factor. U-substitution will work.

Step 2: Choose u. Let u = x² + 1.

Step 3: Find du. du/dx = 2x, so du = 2x dx. This means x dx = du/2.

Step 4: Substitute.
∫ 6x · (x² + 1)² dx = ∫ 6 · u² · x dx = ∫ 6u² · (du/2) = ∫ 3u² du.

Step 5: Integrate.
Apply the power rule to u²: 3 · (u³/3) = u³.

Step 6: Substitute back. Replace u with x² + 1: (x² + 1)³.

Step 7: Add + C.

✅ Answer: (x² + 1)³ + C

Check: Differentiate using the chain rule → 3(x² + 1)² · 2x = 6x(x² + 1)². ✓

Full Solution:

Rule to use: Integration by parts — formula: ∫ u · dv = u · v − ∫ v · du.

Step 1: Identify the structure. This is a product of two different function types — algebraic (x) and exponential (eˣ). Integration by parts applies.

Step 2: Choose u and dv using LIATE.
Algebraic comes before Exponential in LIATE:
u = x  and  dv = eˣ dx.

Step 3: Find du.
Differentiate u = x: du = dx.

Step 4: Find v.
Integrate dv = eˣ dx: v = eˣ.

Step 5: Apply the formula.
∫ x · eˣ dx = x · eˣ − ∫ eˣ · dx.

Step 6: Solve the remaining integral.
∫ eˣ dx = eˣ.

Step 7: Combine and add + C.
x · eˣ − eˣ + C = eˣ(x − 1) + C.

✅ Answer: eˣ(x − 1) + C

Check: Differentiate using the product rule → eˣ(x − 1) + eˣ · 1 = eˣ · x − eˣ + eˣ = xeˣ. ✓

📋 Quick Reference — Rules Covered in This Practice Set

• Problems 1 and 2: Power rule — ∫ xⁿ dx = x^(n+1)/(n+1) + C.
• Problem 3: Sum and difference rule — integrate each term separately.
• Problem 4: Power rule with negative exponents.
• Problem 5: Power rule with fractional exponents — rewrite roots first.
• Problem 6: Natural logarithm rule — ∫ (1/x) dx = ln|x| + C.
• Problem 7: Trig antiderivative rules — ∫ cos(x) dx = sin(x) + C and ∫ sin(x) dx = −cos(x) + C.
• Problem 8: Exponential rule — ∫ e^(ax) dx = (1/a) · e^(ax) + C.
• Problem 9: U-substitution — substitute inner function, cancel, integrate, substitute back.
• Problem 10: Integration by parts — ∫ u · dv = u · v − ∫ v · du using LIATE.

How do I know which rule to use for each antiderivative problem?

Start by identifying the type of function in front of you. If it is a variable with an exponent, reach for the power rule first. If it is 1/x, use the natural log rule. If it is a trig function, apply the matching trig antiderivative rule. If it is eˣ or e^(ax), use the exponential rule. If two different function types are multiplied together, try u-substitution if one is the derivative of the other — or use integration by parts if not. With practice, recognizing the right rule becomes fast and instinctive.

What should I do if my answer looks different from the solution?

Do not immediately assume you are wrong. Two antiderivative expressions can look different but still both be correct if they differ only by a constant. The most reliable test is to differentiate both versions. If both give back the original function, both answers are valid. If your version does not differentiate back to the original function, go through your working line by line to find the exact step where the error occurred — that investigation is where the deepest learning happens.

Can I use an antiderivative calculator to check my answers to these practice problems?

Yes — and it is strongly encouraged as part of your practice routine. After attempting each problem and checking by differentiation, enter the original function into the antiderivative calculator on this site for a final verification. If your answer matches, your understanding of the rule is solid. If it does not, compare the two answers carefully and trace back through your steps to identify the difference. Using the calculator this way turns each practice problem into a complete learning cycle — attempt, verify by hand, confirm with the tool, investigate any differences.