Integration Basics: 6-Mark Paper 1 in 9 Min

1. 2x⁴ − x³ + x + C [3].
2. Write 6x^(−3) [1]; 6x^(−2) ÷ (−2) [1] = −3/x² + C [1].
3. f(x) = 4x^(3/2) + C [2]; 20 = 4 × 8 + C, so C = −12 [1]; f(x) = 4x^(3/2) − 12 [1].

Try them at the end of the video first. Questions are original, in the style of the exam; marks are estimates.

By the end of this video, you will answer this six mark question.

Three ideas decide the marks: integration undoes differentiation, the reverse power rule, and the constant, plus C. It is Paper 1 style, so no calculator.

Two quick questions on differentiation first. Differentiate x cubed.

Three x squared. Multiply by the power, then take one off the power.

Second. Differentiate x squared plus seven.

Two x. The seven disappears. Remember that: it is the reason for plus C.

Idea one. Integration undoes differentiation. Differentiate x cubed, and you get three x squared. So integrate three x squared, and you go back to x cubed. The tall S is the integral sign, and d x tells you the variable.

But look. x cubed plus five also has the derivative three x squared. So does x cubed minus two. The derivative cannot see the constant. So we write x cubed, plus C, where C is any constant. This is called an indefinite integral.

Small task. The integral of two x.

x squared, plus C. Because the derivative of x squared is two x.

Idea two. The reverse power rule. To differentiate, you multiply by the power, then subtract one. To integrate, do the opposite steps in the opposite order: add one to the power, then divide by the new power.

Example. The integral of x to the power four. Add one: the power is five. Divide by five. A number in front just waits: the integral of six x squared is six times x cubed over three, which is two x cubed. And the integral of a constant, like three, is three x.

Careful here. Two classic traps. First, the rule fails when the power is minus one, because you would divide by zero. One over x needs its own rule, in part three. Second, rewrite before you integrate. The square root of x is x to the power one half. One over x squared is x to the power minus two.

Small task. The integral of one over x squared.

Minus one over x, plus C. Write it as x to the power minus two. Add one: the power is minus one. Then divide by minus one.

Idea three. Finding C. Plus C means a whole family of curves, one above the other. One point on the curve picks one member of the family. So if you know the derivative and one point, you can find the exact function.

Example. The derivative of y is two x, and the curve passes through the point one, four. Integrate: y equals x squared plus C. Substitute x equals one and y equals four. Four equals one plus C. So C is three, and y equals x squared plus three.

Careful here. In an indefinite integral, forgetting plus C usually costs the last mark. And when a point is given, finish the job: write the full function, with the value of C in it.

Small task. y equals x squared plus C passes through the point two, ten. Find C.

Six. Ten equals four plus C.

Now three integrals. The method is always the same: rewrite, integrate each term, then add C.

The integral of four x cubed, minus six x, plus five. Go term by term. Four x cubed gives x to the power four. Minus six x gives minus three x squared. Five gives five x. Then plus C. Check it in five seconds: differentiate your answer, and you must get back to the start.

Next. The integral of the square root of x. I rewrite it as x to the power one half. You finish. Add one to the power, then divide.

The new power is three halves. Dividing by three halves is multiplying by two thirds. So the answer is two thirds, x to the power three halves, plus C.

Now alone. The derivative of y is three x squared minus four, and the curve passes through the point two, one. Find y. Pause the video.

Integrate: y equals x cubed, minus four x, plus C. Substitute the point: one equals eight, minus eight, plus C. So C is one, and y equals x cubed, minus four x, plus one.

Now you are the examiner. A three mark question: find the integral of six x squared plus two. Three students answer. Who earns all three marks? You have five seconds.

Student A differentiated instead of integrating, and wrote twelve x. No marks. Student B wrote two x cubed plus two x, and stopped. No plus C: two marks. Student C wrote two x cubed, plus two x, plus C. Three marks.

Indefinite integral? Always plus C.

Back to the question from the start. Pause the video and try both parts.

Part a. Six x squared gives two x cubed. One mark. Minus four x gives minus two x squared. One mark. Three gives three x, and then plus C. One mark.

Part b. f of x is the same integral, with plus C. One mark. Substitute the point one, five: five equals two, minus two, plus three, plus C. One mark. So C is two, and f of x is two x cubed, minus two x squared, plus three x, plus two. One mark.

Six out of six.

Recap. Integration undoes differentiation. The reverse power rule: add one to the power, then divide by the new power. Rewrite roots and fractions as powers first. An indefinite integral always ends with plus C. And one point on the curve gives you C.

Three questions are on screen. Try them today, with no calculator. The answers are in the description.

Next Monday, part two: definite integrals, and the area under a curve. See you there!