Anti-differentiate by recognition from a supplied derivative

The derivative relation is supplied. What should you do first?

Vary k for a primitive of 2e^(kx)

Which conclusion fits every setting?

Anti-differentiate by recognition

When the derivative of a function is supplied, use that relation to identify an anti-derivative directly. If the target is a constant multiple of the supplied derivative, keep the same multiple on the function

1. 1Read the supplied relation A supplied derivative gives an anti-derivative without integrating its visible factors separately \(\displaystyle F'(x)=f(x)\Rightarrow\int f(x)\,dx=F(x)+C\)
2. 2Carry a constant scale A constant multiplying the derivative also multiplies the recognised function \(\displaystyle g(x)=2F'(x)\Rightarrow\int g(x)\,dx=2F(x)+C\)
3. 3Use one additive constant The complete family has one arbitrary constant for the whole anti-derivative \(\displaystyle \int\left(f(x)+h(x)\right)\,dx=F(x)+H(x)+C\)
4. 4Differentiate to check Differentiate the proposed primitive and compare it with the target function \(\displaystyle \frac{d}{dx}\left(2F(x)+C\right)=2F'(x)\)

Given

\(\displaystyle \frac{d}{dx}\left(2\left(x^2+3\right)^{3/2}\right)=\square x\sqrt{x^2+3}\)

Differentiate the supplied function and find the coefficient of the displayed derivative term

Given

\(\displaystyle g(x)=3F'(x)\Rightarrow\int g(x)\,dx=\square F(x)+C\)

Use the scale between the target function and the supplied derivative

Using the derivative as the anti-derivative. One of these lines is right. Which one?

Construct the whole primitive family

For a nonzero linear inner coefficient a, divide by a when reversing the chain rule. Power n≠−1 also gains exponent n+1 and divisor n+1. Differentiate the complete candidate to check every term and retain one arbitrary constant

Construct the full expression

Construct the complete family without a supplied candidate. Include the cosine sign, all inner divisors and C. Differentiate your expression and match every term

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle \int[8(2x-1)^3-6e^{-2x}+5\cos5x]dx\)

What coefficient multiplies e^(−2x) in your primitive?

Review the whole expression

The power term has coefficient 8/(2×4)=1; the exponential term has coefficient 3; the cosine term integrates to sin5x. All are defined on R; C represents the family

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

Handle the exceptional reciprocal power

The power formula cannot divide by n+1 when n=−1. Instead use a logarithm with an absolute value on each connected interval excluding the pole. Different constants may be chosen on disconnected intervals

Construct the full expression

Find the complete primitive on each interval of its domain, retaining the absolute value. Verify by differentiation and explain why a constant must not be assumed shared across a pole

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle \int\left[\frac5{2x+1}+3\sqrt{2x+4}\right]dx\)

What coefficient multiplies ln|2x+1|?

Review the whole expression

The integrand domain is x≥−2 excluding −1/2. On the open intervals (−2,−1/2) and (−1/2,∞), differentiate the formula to recover the integrand. It extends continuously from the right to −2; an ordinary two-sided derivative is claimed only inside the domain. Constants can differ between the two components

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

Construct a recognised primitive and its constant

Differentiate a promising whole expression and compare it with the target. The derivative of e^(x²) is 2xe^(x²), so the target 6xe^(x²) is three times that derivative. The condition at zero then fixes C.

Construct the full expression

Recognise a complete primitive, derive its scale factor and apply the initial condition. Show the product/chain derivative verification; do not integrate the two visible factors separately

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle G^{\prime}(x)=8x(x^2+1)^3,\qquad G(0)=4\)

After finding the primitive, give its additive constant C.

Review the whole expression

Differentiate (x²+1)^4: 4(x²+1)^3×2x gives exactly the target. The initial condition yields C=3; the resulting function and derivative have domain R.

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression